Cajori
History of mathematics
48 00376 4479
MA
MATHEMATICS
BY
PLORIAN CAJORI
FOHMEBLY PROFESSOR 03T APPLIED MATHEMATICS IN THE TULANE UNIVERSITY
OF LOUISIANA; NOW PROCESSOR or PHYSICS
IN COLORADO COLLEGE
* I am sure that no subject loses more than mathematics
"by any attempt to dissociate it from, its history." J. W. L.
GLAISHMR
gatfc
MACMILLAN AND CO.
ANB LONDON
1894
All riff Ms reserved
COPYRIGHT, 189,1,
BY MAOM1LLAN AND 00,
*
ITortooob prt ;
J. S, Gushing & Co, -Berwick & Smith,
Boston, Mass,, U.S.A.
PREFACE.
AN increased interest in the history of the exact sciences
manifested in recent years by teachers everywhere, and the
attention given to historical inquiry in the mathematical
class-rooms and seminaries of our leading universities, cause
me to believe that a brief general History of Mathematics
will be found acceptable to teachers and students.
The pages treating necessarily in a very condensed
form of the progress made during the present century,
are put forth with great diffidence, although I have spent
much time in the effort to render them accurate and
reasonably complete. Many valuable suggestions and criti
cisms on the chapter on "B/ecent Times" have been made
by ,I)r. E. W. Davis, of the University of Nebraska. The
proof-shoots o f this chapter have also been submitted to
Dr. J. E. Davies and Professor C. A. Van Velzer, both of the
University of Wisconsin; to Dr. G-. B. Halsted, of the
University of Texas ; Professor L. M. HosMns, of the Leland
Stanford Jr. University ; and Professor Gr. D. Olds, of Amherst
College, all of whom have afforded valuable assistance.
1 am specially indebted to Professor 3T. H. Loud, of Colorado
College, who has read the proof-sheets throughout. To all
the gentlemen above named, as well as to Dr. Carlo Veneziani
v
vi PKEFACE.
of Salt Lake City, who read the first part of my work in.
manuscript, I desire to express my hearty thanks. But in
acknowledging their kindness, I trust that I shall not seem
to lay upon them any share in the responsibility for errors
which I may have introduced in subsequent revision of the
FLORIAN CAJOBL
COLORADO COLLEGE, December, 1893.
TABLE OF CONTENTS.
PAGE
INTRODUCTION 1
, ANTIQUITY 5
THE BABYLONIANS 5
THE EGYPTIANS 9
THE GREEKS 16
Greek Geometry 16
The Ionic School 17
The School of Pythagoras 19
The Sophist School 23
The Platonic School 29
The First Alexandrian School 34
The Second Alexandrian School 54
Greek Arithmetic 63
TUB ROMANS 77
^ MIDDLE AGES 84
THE HINDOOS 84
THE ARABS 100
EtJBOPE DURING THE MIDDLE AOES 117
Introduction of Roman Mathematics 117
Translation of Arabic Manuscripts 124
The First Awakening and its Sequel 128
MODERN EUROPE 138
THE RENAISSANCE : . . . . 189
VIETA TO DJCSOARTES ^
DBSGARTES TO NEWTON 183
NBWTGN TO EULBK 199
vii
Viii TABLE OF CONTENTS.
PAGE
EULER, LAGRANGE, AND LAPLACE 246
The Origin of Modern Geometry 285
KECENT TIMES 291
SYNTHETIC GEOMETRY 293
ANALYTIC GEOMETRY 307
ALGEBRA 315
ANALYSIS 331
THEORY OP FUNCTIONS 347
THEORY OF NUMBERS 362
APPLIED MATHEMATICS 373
INDEX 405
BOOKS OF REFEKENCE.
The following books, pamphlets, and articles have been used
in the preparation of this history. Reference to any of them
is made in the text by giving the respective number. Histories
marked with a star are the only ones of which extensive use
has been made.
1. GUNTHER, S. Ziele tmd Hesultate der neueren Mathematisch-his-
torischen JForschung. Erlangen, 1876.
2. CAJTOEI, F. The Teaching and History of Mathematics in the U. S.
Washington, 1890.
3. *CANToit, MORITZ. Vorlesungen uber Gfeschichte der MathematiJc.
Leipzig. Bel I., 1880; Bd. II., 1892.
4. EPPING, J. Astronomisches aus Babylon. Unter Mitwirlcung von
P. J. K. STUASSMAIER. Freiburg, 1889.
5. BituTHOHNKiDfflR, C. A. Die Qeometrie und die G-eometer vor Eukli-
des. Leipzig, 1870.
6. * Gow, JAMES. A Short History of Greek Mathematics. Cambridge,
1884.
7. * HANKBL, HERMANN. Zur Gfeschichte der MathematiJc im Alterthum
und Mittelalter. Leipzig, 1874.
8. *ALLMAN, G. J. G-reek G-eometr y from Thales to JEuclid. Dublin,
1889.
9. DB MORGAN, A. "Euclides" in Smith s Dictionary of Greek and
Itoman Biography and Mythology.
10. HANKBL, HERMANN. Theorie der Complexen Zahlensysteme. Leip
zig, 1807.
11. WmcwELL, WILLIAM. History of the Inductive Sciences.
12. XEUTIIISN, II. G. Die Lehre von den Kegelschnitten im Alterthum.
KopQnlaagen, 1886.
ix
X BOOKS OF REFERENCE.
13. * CHASLES, M. G-eschichte der Geometric. Aus dem Franzosischen
tibertragen durcli DR. L. A. SOHNCKE. Halle, 1839.
14. MARIE, MAXIMILIEN. Histoire des Sciences Matheniatiques et Phy
siques. Tome I.-XII. Paris, 1883-1888.
15. COMTE, A. Philosophy of Mathematics, translated by W. M. GIL-
LESPIE.
16. HANKEL, HERMANN. Die ISntwickelung der Mathematik in den letz-
ten Jahrhunderten. Tubingen, 1884.
17. GUNTHER, SIEGMUND und WiNBELBAND, W. GesckicJite der antiJcen
Naturwissenschaft und Philosophic. Nordlingen, 1888.
18. ARNETH, A. Geschichte der reinen Mathematik. Stuttgart, 1852.
19. CANTOR, MOIUTZ. Mathematische Beitrage zum Kulturleben der
VoUcer. Halle, 1863.
20. MATTIIIESSEN, LTIDWIG. Grundzilge der Antiken und Modernen
Algebra der Litteralen GUichungen. Leipzig, 1878.
21. OURTMANN und MULLER. Fort$chritte der Mathematik.
22. PEACOCK, GEORGE. Article " Aritlimetic, 1 in The Encyclopedia, of
Pure Mathematics. London, 1847.
23. HERSCHEL, J. !F. W. Article * Mathematics," in Edinburgh Jfflncy-
dopcedia.
24. SUTER, HEINRICH. Cfeschichte der Mathematischen Wissenschaften.
Zurich, 1873-75.
25. QUETELET, A. Sciences Mathetna&iques et IViysiques ehe% les Beiges.
Bruxelles, 1866.
26. PLAYFAIR, JOHN. Article u Progress of the Mathematical and Phys
ical Sciences," in Encyclopedia Britannica, 7th editi6n, con
tinued in the 8tlx edition by SIK JOHN LESLIE.
27. BE MORGAN, A. Arithmetical Books from the Invention of Printing
to the Present Time.
28. NAPIER, MARK. Memoirs of John Napier of Merchiston. Edin
burgh, 1834.
29. HALSTEB, G. B. "Note on the First English Euclid," American
Journal of Mathematics, Vol. XL, 1879.
30. MADAME PERIER. The Life of Mr. Paschal. Translated into
English by W. A., London, 1744.
31. MONTUCLA, J. F. Histoire des Mathematiques. Paps, 1802.
32. BtiHRiNG- E. Kritische Geschichte der allgemeimn Principien der
Mechanik. Leipzig, 1887.
33. BREWSTER, D. The Memoirs of Nc.wton. Edinburgh, 1860.
^81. BALL, W. W. R. A Short Account of the History of Mathematics.
London, 1888, 2nd edition, 189S,
35. DE MORGAN, A. "On the Early History of lEfiEitesixualB," in the
Philosophical Magazine, November, 1852.
BOOKS OF REFERENCE, xi
36. Bibliotheca Mathematica, herausgegeben von GUSTAP ENESTROM,
Stockholm.
37. GUNTHER, SIEGMUND. Vermischte Untersuchtingen zur Geschichte
der mathematischen Wissenschaften. Leipzig, 1876.
38. *GERHARDT, C. I. Geschichte der Mathematik in, Deutschland.
Miinclien, 1877.
39. GERHARDT, C. I. SntdecJcung der Di/erenzialrechnung durch Leib
niz. Halle, 1848.
40. GERIIARBT, K. I. " Leibniz in London," in jSitzirngsberichte der
Koniglich Preussischen Academic der Wissenschaften zu Berlin,
FeTbruar, 1891.
41. DB MoR(UK, A. Articles "Muxions" and u Commercimn Epistoli-
cum," in tlie Penny Cyclopaedia,
42. *TODUUNTEK, I. A History of the Mathematical Theory of Probabil
ity from the Time of Pascal to that of Laplace. Cambridge and
London, 1865.
43. *Toi>iniNTBK, I. A History of the Theory of Elasticity and of the
Strength of Materials. Edited and completed by KARL PEARSON.
Cambridge, 1886.
44. TOBHUNTKR, I. " Note on tlie History of Certain Formulas in Spher
ical Trigonometry," Philosophical Magazine, February, 1873.
46. Die JBasler Mathematiker, Daniel Bernoulli und Leonhard Euler.
BaHol, 1884,
46. RKIFF, R. Gfeschichte der Unendlichen Heihen. Tubingen, 1889.
47. WALTKRSIIAUSKN , W. SAHTOUIXIS. Gauss , mm Q-ed&chniss. Leip
zig, 1850.
48. BAUMCJART, OSWALO. Ueber das Quadratische J&eciprocitatsgesetz,
Leipzig, 1885.
49. HATHAWAY, A. S. "Early History of the Potential," Bulletin of
the N. Y. Mathematical Society, I. 3.
50. WOLF, RUDOLF. Cfeschichte der Astronomie. Mtinchen, 1887.
51. AUAOO, 1). F. J. " Eulogy on Laplace. 7 Translated by B. POWELL,
Smitlisonian llPfiort, 1874,
52* BEAUMONT, M. I^LIK DB. "Memoir of Legendre." Translated by
C. A. ALEXANDISR, Smithsonian Iteport, 1867.
58. AUAOO, I). F. X Joseph Fourier." Smithsonian Eeport,
1871.
54, WITHER, CnuiHTiAN. Lehrfatch der Darstellenden Gfeometrie. Leip
zig, 1884.
55. *LoiA, GTKO. Die Ilmptsilchliehstm Theorien der Geometrie in
ihrer fr dhtren und heutlgen fJntwicMnnff, ins deutsche tibertra-
gen von Fitm SOHUTTB. Leipzig, 1888.
Xll BOOKS OF REFERENCE.
56 . CAYLE Y, ARTHUR. Inaugural Address before the British Association,
1883.
57. SPOTTISWOODE, WILLIAM. Inaugural Address before the British
Association, 1878.
58. GIBBS, J. WILLARD. " Multiple Algebra," Proceedings of the
American Association for the Advancement of Science, 1886.
59. FINK, KARL. Geschichte der Elenientar-Mathematik. Tubingen,
1890.
60. WITTSTEIN, ARMIN. Zur Qeschichte des Malfatttf schen Problems.
Nordlingen, 1878.
61. KLEIN, FELIX. Vergleichende Betrachtimgen uber neuere geome-
trische Forschungen. Erlangen, 1872.
62. FORSYTH, A. R. Theory of Functions of a Complex Variable.
Cambridge, 1893.
63. GRAHAM, R. H. Geometry of Position. London, 1891.
64. SCHMIDT, FRANZ. "Aus dem Leben zweier ungarischer Mathe-
matiker Johann und Wolfgang Bolyai von Bolya." Grrunertfs
Archiv, 48:2, 1868.
65. FAVARO, ANTON. Justus Bellavitis," Zeitschrift fur Mathematik
und Physik, 26 : 5, 1881.
66. BRONICE, AD. Julius Plucker. Bonn, 1871.
67. BAUER, GUSTAV. Gfedachnissrede auf Otto Hesse. Miinchen,
1882.
68. ALFRED CLEBSCH. Versuch einer Darlegung und Wunligung seiner
wissenschaftlichen Leistungen von einigen seiner Freunde. Leip
zig, 1873.
69. HAAS, AUGUST. Versuch einer Darstellung der Geschichte dvs
Krwnmungsmasses. Tubingen, 1881.
70. FINE, HENRY B. The Number- System of Algebra. Boston and
New York, 1890.
71. SCHLEGEL, VICTOR. Hermann Gfrassmann, sein Leben und seine
Werke. Leipzig, 1878.
72. ZAHN, W. v. " Einige Worte zum Andenkon an Hermann Ilankol,"
Mathematische Annalen, VII. 4, 1874.
73. MUIR, THOMAS. ^1 Treatise on Determinants*. 1882.
74. SALMON, GEORGE. "Arthur Cayley," Nature, 28:21, September,
1883.
75. CAYLEY, A. "James Joseph Sylvester," Nature, 39:10, January,
1889.
76. BURKHARDT, HEiNRicii. Die AnfUngo der Gruppontliooiie und
Paolo Ikiffim," Zeitschrift der MathemaUk und Physik, Supple
ment, 1892.
BOOKS OF REFERENCE. xiii
77. SYLVESTER , J. J. Inaugural Presidential Address to the Mathe
matical and Physical Section of the British Association at Exeter.
1869.
78. YALSON, C. A. La Vie et les travaux du Baron Cauchy. Tome I.,
II., Paris, 1868.
79. SACHSE, ARNOLD. Versuch einer Qeschichte der Darstellung will-
kiirlicher Funktionen einer variablen durch trigonometrische
Meihen. Gottingen, 1879.
80. BOIS-KEYMOND, PAUL DU. Zur G-eschichte der Trigonometrischen
Heilien, Mine JBntgegnung. Tubingen.
81. POINCARE, HENRI. Notice sur les Travaux Scientifiques de Henri
Poincare. Paris, 1886.
82. BJERKNES, C. A. Niels-HenriTc Abel, Tableau de sa vie et de son
action scientifique. Paris, 1885.
83. TUCKER, R. "Carl Friedrich Gauss," Nature, April, 1877.
84. DIRICHLET, LEJEUNE. Gfedachnissrede auf Carl Gf-iistav Jacob
Jacobi. 1852.
85. ENNEPER, ALFRED. JUlliptische JFunktionen. Theorie und Ge-
schichte. Halle a/S., 1876.
86. HENRICI, O. "Theory of Functions," Nature, 43 : 14 and 15, 1891.
87. DARBOUX, GASTON. Notice stir les Travaux Scientijlques de M. Gas-
ton Darboux. Paris, 1884.
88. KUMMER, E. E. Gfedachnissrede auf G-ustav Peter Lejeune-Dirichlet.
Berlin, 1860.
89. SMITH, H. J. STEPHEN. "On the Present State and Prospects of
Some Branches of Pure Mathematics," Proceedings of the London
Mathematical Society, Vol. VIII, Nos. 104, 105, 1876.
90. GLAISUISH, J. W. L. " Henry John Stephen Smith, Monthly Notices
of the Eoyal Astronomical Society, XLIV., 4, 1884.
91. Bessel als Bremer Ifandlungslehrling. Bremen, 1890.
92. FRANTZ, J*. Festrede aus Veranlassung von HesseVs hundertjahrigem
Geburtstag. Konigsherg, 1884.
93. DZIOBEK, 0. Mathematical Theories of Planetary Motions.
Translated into English by M. "W". Harrington and "W. J. Hussey.
94. HERMITB, Cn. "Discours prononc6 devant le president de la R6pii-
Tblique," Bulletin des Sciences Mathematiques, XIV., Janvier,
1890.
95. SCHUSTER, ARTHUR. "The Influence of Mathematics on the Prog
ress of Physics," Nature, 26: 17, 1882.
96. KERBEDJS, E. BE. "Sophie de KowalevsM," Bendiconti del Circolo
Matematico di Palermo, V., 1891.
,97. VOIGT, W. Zum Gfeddchniss von G. Kirchhoff. Gottingen, 1888.
xiv BOOKS O:F BEFBBBNCE,
08) Bdci-iER, MAXIME. u A Bit of Mathematical History," Bulletin of
the 2V". T. Math. /SV>c., Vol. II., No. 5.
99. CAY:LEY, ARTHUR. Report on the Recent Progress of Theoretical
Dynamics. 1857.
100. GLAZEBROOK, U. T. Report on Optical Theories. 1885.
101. ROSENBERGER, If. Geschichte tier Physik. Braunschweig, 1887-1890.
A HISTORY OF MATHEMATICS.
INTEODUCTION.
THE contemplation of the various steps by which mankind
has come into possession of the vast stock of mathematical
knowledge can hardly fail to interest the mathematician. He
takes pride in the fact that his science, more than any other,
is an exact science, and that hardly anything ever done in
jnatheBiati.es has proved to be useless. The chemist smiles
at the childish, efforts of alchemists, but the mathematician
finds the geometry of the Greeks and the arithmetic of the
Hindoos as useful and admirable as any research of to-day.
He is pleased to notice that though, in course of its develop
ment, mathematics has had periods of slow growth, yet in
the main it has been pre-eminently a progressive science.
The history of mathematics may be instructive as well as
agreeable 5 it may not only remind us of what we have, but
inay also teach us how to increase our store. Says De Morgan,
* The early history of the mind of men with regard to mathe
matics leads us to point out our own errors; and in this
* aspect it is well to pay attention to the history of mathe
matics." It warns us against hasty conclusions ; it points out
the importance of a good notation upon the progress of the
science ; it discourages excessive specialisation on the part of
1
2 A HISTORY OF MATHEMATICS.
investigators, by showing how apparently distinct brandies
have been found to possess unexpected connecting links; it
saves the student from wasting time and energy upon prob
lems which were, perhaps, solved long since; it discourages
him from attacking an unsolved problem by the same method
which has led other mathematicians to failure ; it teaches that
fortifications can be taken in other ways than by direct attack,
that when repulsed from a direct assault it is well to recon
noitre and occupy the surrounding ground and to discover the
secret paths by which the apparently unconquerable position
can be taken. 1 The importance of this strategic rule may
be emphasised by citing a case in which it has been violated.
(An untold amount of intellectual energy has been expended
on the quadrature of the circle, yet no conquest has been made
by direct assault. The circle-squarers have existed in crowds
ever since the period of Archimedes. After innumerable fail
ures to solve the problem at a time, even, when investigators
possessed that most powerful tool, the differential calculus,
persons versed in mathematics dropped the subject, wMlo
those who still persisted were completely ignorant of its Ms-
tory and generally misunderstood the conditions of the prob
lem.^ "Our problem," says De Morgan, "is to square the
circle with the old allowance of means: Euclid s postulates
and nothing more. We cannot remember an instance tyx
a question to be solved by a definite method was tried by\$k6
best heads, and answered at last, by that method, after thou
sands of complete failures." But progress was made on this
problem by approaching it from a different direction and by
newly discovered paths. Lambert proved in 1761 that
ratio of the circumference of a circle to its diametot is iad0.ni-
meiisurable. Some years ago, Linclomaim demonstrated that
this ratio is also transcendental and that the quadrature <>
the circle, by means of the ruler and compass only, is
INTBODUCTION. 3
sible. He thus showed by actual proof that which keen-
minded mathematicians had long suspected ; namely, that the
great army of circle-squarers have, for two thousand years,
been assaulting a fortification which is as indestructible as
the firmament of heaven.
Another reason for the desirability of historical study is
the value of historical knowledge to the teacher of mathe
matics. The interest which pupils take in their studies may
bo greatly increased if the solution of problems and the cold
logic of geometrical demonstrations arc interspersed with
historical remarks and anecdotes. A class in arithmetic will
be pleased to hear about the Hindoos and their invention of
the " Arabic notation " ; they will marvel at the thousands
of years which elapsed before people had even thought of
introducing into the numeral notation that Coluni bus-egg
the zero j they will find it astounding that it should have
taken so long to invent a notation which they themselves can
now learn in a month. After the pupils have learned how to
bisect a given angle, surprise them by telling of the many
futile attempts which have been made to solve, by elementary
geometry, the apparently very simple problem of the trisec-
tion of an angle. When they know how to construct a square
whose area is double the area of a given square, tell them
about the duplication of the cube how the wrath of ^Apollo
could be appeased only by the construction of a cubical altar
double the given altar, and how mathematicians long wrestled
with this problem. After the class have exhausted their ener
gies on the theorem of the right triangle, tell them something
about its discoverer how Pythagoras, jubilant over his great
accomplishment, sacrificed a hecatomb to the Muses who in-
him. When the value of mathematical training is
in question, quote the inscription over the entrance into
i academy of Plato, the philosopher : " Let no one who is
4 A HISTORY OF MATHEMATICS.
unacquainted with geometry enter here." Students in analyt^
ical geometry should know something of Descartes, and, after
taking up the differential and integral calculus, they should
become familiar with the parts that Kewton, Leibniz, and
Lagrange played in creating that science. In his historical
talk it is possible for the teacher to make it plain to the
student that mathematics is not a dead science, but a living
one in which steady progress is made. 2
The history of mathematics is important also as a valuable
contribution to the history of civilisation. Human progress
is closely identified with scientific thought. Mathematical
and physical researches are a reliable record of intellectual
progress. The history of mathematics is one of the large
windows through which the philosophic eye looks into past
ages and traces the line of intellectual development.
ANTIQUITY,
THE BABYLONIANS,
THE fertile valley of the Euphrates and Tigris was one of
the primeval seats of human society. Authentic history of
the peoples inhabiting this region begins only with the foun
dation, in Chaldaoa and Babylonia, of a united kingdom out
of tho previously disunited tribes. Much light has been
thrown on their history by the discovery of the art of reading
the cuneiform or wedge-shaped system of writing.
In the study of Babylonian mathematics we begin with the
notation of numbers, A vertical wedge If stood for 1, while
the I characters" ^ and y>*. signified 10 and 100 respec
tively. G-rotefend believes the character for 10 originally to
been the picture of two hands, as held in prayer, the
palniis being pressed together, the fingers close to each other,
btiTOhe thumbs thrust out, In the Babylonian notation two
ptincjiiples were employed the ^ditive) and multiplica
tive. |i Numbers below 100 were expressed by symbols whose
respt-Mctive values had to be added. ^ Thus, y stood for 2,
|f )f | |or 3, XJJ1 for 4, <* for 23, ^ ^ < for 30 J Here the
of higher order appear always to the left of those of
I order. In writing the hundreds, on the other hand, a
Ir symbol was placed to the left of the 100, and was, in
fjase, to be multiplied by 100. Thus, s y ^^ signified
the eaf 5
6 A HISTOKY OF MATHEMATICS.
10 times 100, or 1000. But this symbol for 1000 was itself
taken for a new unit, which could take smaller coefficients to
its left. Thus, ^ ^ f >*" denoted, not 20 times 100, but
10 times 1000. Of the largest numbers written in cuneiform
symbols, which have hitherto been found, none go as high as
a million. 3
If, as is believed by most specialists, the early Sumerians
were the inventors of the cuneiform writing, then they were,
in all probability, also familiar with the notation of numbers.
Most surprising, in this connection, is the fact that Sumerian
inscriptions disclose the use, not only of the above decimal
system, biit also of a sexagesimal one. The latter was used
chiefly in constructing tables for weights and measures. It is
full of historical interest. Its consequential development,
both for integers and fractions, reveals a high degree of
mathematical insight. We possess two Babylonian tablets
which exhibit its use. One of them, probably written between
2300 and 1600 B.C., contains a table of square numbers up to
601 The numbers 1, 4, 9, 16, ,25, 36, 49, are given as v the
squares of the first seven integers respectively. We have next
1.4 = 8 s , 1.21 = 9 2 , 1.40 = 10 2 , 2.1 = 11*, etc. This"reinLfta
unintelligible, taxless we assume the sexagesimal scale, wl xioh
makes 1.4 = 60 + 4, 1.21 = 60 + 21, 2.1 = 2.60 + 1. The i
tablet records the magnitude of the illuminated portion of
moon s disc for every day from new to full moon, the wlxol
being assumed to consist of 240 parts, The illuminated
during the first five days are the series 5, 10/ 20, 40,
(=80), which is a geometrical progression. From
the series becomes an arithmetical progression, the
from the fifth to the fifteenth day being respectively 1,20!
1.62, 2.8, 2.24, 2.40, 2.66, 3.12, 3.28, 3,44, 4. This
only exhibits the use of the sexagesimal system, but
cates the acquaintance of the Babylonians with
THE BABYLONIANS. 7
Not to be overlooked is the fact that in the sexagesimal nota-.
tion of integers the "principle of position" was employed.
Thus, in 1.4 (=64), the 1 is made to stand for 60, the unit
of the second order, by virtue of its position with respect to
the 4. The introduction of this principle at so early a date
is the more remarkable, because in the decimal notation it
was not introduced till about the fifth or sixth century after
Christ. The principle of position, in its general and syste
matic application, requires a symbol for zero. We ask, Did
the Babylonians possess one? Had they already taken the
gigantic step of representing by a symbol the absence of
units? Neither of the above tables answers this question,
for they happen to contain no number in which there was
occasion to use a zero. The sexagesimal system was used also
in fractions. Thus, in the Babylonian inscriptions, | and |
are designated by 30 and 20, the reader being expected, in
his mind, to supply the word " sixtieths." The Greek geom
eter Hypsicles and the Alexandrian astronomer Ptolemaeus
borrowed the sexagesimal notation of fractions from the
Babylonians and introduced it into Greece. From that time
sexagesimal fractions held almost full sway in astronomical
and mathematical calculations until the sixteenth century,
when they finally yielded their place to the decimal fractions.
It may be asked, What led to the invention of the sexagesi
mal system ? Why was it that 60 parts were selected ? To
this we have no positive answer. Ten was chosen, in the
decimal system, because it represents the number of fingers.
But nothing of the human body could have suggested 60.
Cantor offers the following theory : At first the Babylonians
reckoned the year at 360 days. This led to the division of
the circle into 360 degrees, each degree representing the daily
amount of the supposed yearly revolution of the sun around
the earth. Now they were, very probably, familiar with the
8 A HISTOKY OF MATHEMATICS.
fact that the radius can be applied to its cir%umference as a
chord 6 times, and that each of these chords subtends an arc
measuring exactly 60 degrees. Fixing their attention upon
these degrees, the division into 60 parts may have suggested
itself tp them. Thus, when greater precision necessitated a
subdivision of the degree, it was partitioned into 60 minutes.
In this way the sexagesimal notation may have originated.
The division of the day into 24 hours, and of the hour
into minutes and seconds on the scale of 60, is due to the
Babylonians.
It appears that the people in the Tigro-Exiphrates basin had
made very creditable advance in arithmetic. Their knowledge
of arithmetical and geometrical progressions has already been
alluded to. lamblichus attributes to them also a knowledge
of proportion, and even the invention of the so-called musical
proportion. Though we possess- no conclusive proof, we have
nevertheless reason to believe that in practical calculation
they used the abacus. Among the races of middle Asia, even
as far as China, the abacus is as old as fable. Now, Babylon,
was once a great commercial centre, the -metropolis of many
nations, and it is, therefore, not unreasonable to suppose that
her merchants employed this most improved aid to calculation,
In geometry the Babylonians accomplished almost nothing.
Besides the division of the circumference into 6 parts by its
radius, and into 360 degrees, they had some knowledge of
geometrical figures, such as the triangle and quadrangle, which
they used in their auguries. Like the Hebrews (1 Kin, 7 : 23),
they took w = 3. Of geometrical demonstrations there is^ of
course, no trace. "As a rule, in the Oriental mind the intui
tive powers eclipse the severely rational and logical."
The astronomy of the Babylonians has attracted much
attention. They worshipped the heavenly bodies from tie
earliest historic times, When Alexander the Great, after
THE EGYPTIANS. 9
the battle of Arbela (331 B.C.), took possession of Babylon,
Callisthenes found there on burned brick astronomical records
reaching back as far as 2234 B.C. Porphyrius says that these
were sent to Aristotle. Ptolemy, the Alexandrian astrono
mer, possessed a Babylonian record of eclipses going back to
747 B.C. Eecently Epping and Strassmaier 4 threw considera
ble light on Babylonian chronology and astronomy by explain
ing two calendars of the years 123 B.C. and 111 B.C., taken
from cuneiform tablets coining, presumably, from an old
observatory. These scholars have succeeded in giving an
account of the Babylonian calculation of the new and full
moon, and have identified by calculations the Babylonian
names of the planets, and of the twelve zodiacal signs and
twenty-eight normal stars which correspond to some extent
with the twenty-eight naksJiatras of the Hindoos. We append
part of an Assyrian astronomical report, as translated by
Oppert :
"To the King, my lord, thy faithful servant, Mar-Istar."
" . . . On the first day, as the new moon s day of the month Tham-
muz declined, the moon was again visible over the planet Mercury, as
I had already predicted to my master the King, I erred not."
THE EGYPTIANS.
Though there is great difference of opinion regarding the
antiquity of Egyptian civilisation, yet all authorities agree in
the statement that, however far back they go, they find no
uncivilised state of society. " Menes, the first king, changes
the course of the Wile, makes a great reservoir, and builds the
temple of Phthah at Memphis." The Egyptians built the
pyramids at a very early period. Surely a people engaging in
10 A HISTOBY OF MATHEMATICS.
enterprises of such magnitude must have known something of
mathematics at least of practical mathematics.
All Greek writers are unanimous in ascribing, without
envy, to Egypt the priority of invention in the mathematical
sciences. Plato in Pho&drus says : " At the Egyptian city
of Naucratis there was a famous old god whoso name was
Theuth; the bird which is called the Ibis was sacred to
him, and he was the inventor of many arts, such as arithmetic
and calculation and geometry and astronomy and draughts
and dice, but his great discovery was the use of letters/ 15
Aristotle says that mathematics had its birth in Egypt,
because there the priestly class had the leisure needful for
the study of it. Geometry, in particular, is said by Herodotus,
Diodorus, Diogenes Laertius, lamblichus, and other ancient
writers to have originated in Egypt. 5 In Herodotus wo find
this (II. c. 109) : " They said also that this king [Sesostjris]
divided the land among all Egyptians so as to give each 0110 a
quadrangle of equal size and to draw from each Ms revenues,
by imposing a tax to be levied yearly. But every one from
whose part the river tore away anything, had to go to hh^i
and notify what had happened; lie then sent the overseers,
who had to measure out by how much th0 land lxad> become
smaller,, in order that the owner might pay on what was left,
in/ proportion to the entire tax imposed, Iti this wny/ifc
appears to me, geometry originated, which passed thence to
Hellas." "-y
We abstain from introducing additional Greek opinion
regarding Egyptian mathematics, or from indulging in wild
conjectures. We rest our account on documentary evidottc^.
A hieratic papyrus, included in the Rhine! collection of tha
British Museum, was deciphered by Eisenlohr in 1877, and
found to be a mathematical manual containing problems in
arithmetic and geometry. It was written by Ataw
THE GREEKS. 17
left behind no written records of their discoveries. A full
jdstory of Greek geometry and astronomy during this period,
written by Eudenus, a pupil of Aristotle, has been lost. It
was well known to Proclus, who, in his commentaries on
Euclid, gives a brief account of it. This abstract constitutes
our most reliable information. We shall quote it frequently
under the name of Eudemian Summary.
The Ionic School
To Thales of Miletus (640-546 B.C.), one of the "seven wise
men," and the founder of the Ionic school, falls the honour of
having introduced the study of geometry into Greece. During
middle life he engaged in commercial pursuits, which took
him to Egypt. He is said to have resided there, and to have
studied the physical sciences and mathematics with the Egyp
tian priests. Plutarch declares that Thales soon excelled his
masters, and amazed King Amasis by measuring the heights
of the pyramids from their shadows. According to Plutarch,
this was dono by considering that the shadow cast by a verti
cal staff of known length bears the same ratio to the shadow
of the pyramid as the height of the staff bears to the height
of the pyramid. This solution presupposes a knowledge of
proportion ? and the Ahmes papyrus actually shows that the
rudiments of proportion were known to the Egyptians. Ac
cording to Diogenes Laertius, the pyramids were measured by
Thales in. a different way ; viz. by finding the length of the
shadow of the pyramid at the moment when the shadow of a
staff was 0(jual to its own length.
The JSud&mian Summary ascribes to Thales the invention
of the theorems on the equality of vertical angles, the equality
af the angles at the base of an isosceles triangle, the bisec
tion of a circle by any diameter, and the congruence of two
18 A HISTORY OF MATHEMATICS.
triangles having a side and the two adjacent angles equal re
spectively. The last theorem he applied to the measurement
of the distances of ships from the shore. Thus Thales was
the first to apply theoretical geometry to practical uses. The
theorem that all angles inscribed in a semicircle are right
angles is attributed by some ancient writers to Thales, by
others to Pythagoras. Thales was doubtless familiar with
other theorems, not recorded by the ancients. It has been
inferred that he knew the sum of the three angles of a tri
angle to be equal to two right angles, and the sides of equi
angular triangles to be proportional. 8 The Egyptians must
have made use of the above theorems on the straight line, in
some of their constructions found in the Ahmes papyrus, but
it was left for the Greek philosopher to give these truths,
which others saw, but did not formulate into words, an
explicit, abstract expression, and to put into scientific lan
guage and subject to proof that which others merely felt to
be true. Thales may be said to have created the geometry
of lines, essentially abstract in its character, while the. Egyp
tians studied only the geometry of surfaces and the rudiments
of solid geometry, empirical in their character. 8
With Thales begins also the study of scientific astronomy.
He acquired great celebrity by the prediction of a solar eclipse
in 585 B.C. Whether he predicted the day of the occurrence,
or simply the year, is not known. It is told of him that
while contemplating the stars during an evening walk, he fell
into a ditch. The good old woman attending him exclaimed,
"How canst thou know what is doing in the heavens, when
thou seest not what is at thy feet ? "
The two most prominent pupils of Thales were Anaximander
(b. 611 B.C.) and Anaximenes (b. 570 B.C.). They studied
chiefly astronomy and physical philosophy. Of Anaxagoras, ;a
pupil of Anaximenes, and the last philosopher of the Ionic
THE GREEKS. 19
school, we know little, except that, while in prison, he passed
his time attempting to square the circle. This is the first
time, in the history of mathematics, that we find mention of
the famous problem of the quadrature of the circle, that rock
upon which so many reputations have been destroyed. It
turns upon the determination of the exact value of IT. Approx
imations to TT had been made by the Chinese, Babylonians,
Hebrews, and Egyptians. But the invention of a method to
find its exact value, is the knotty problem which has engaged
the attention of many minds from the time of Anaxagoras
down to our own. Anaxagoras did not of er any solution of
it, und seems to have luckily escaped paralogisms.
About the time of Anaxagoras, but isolated from the Ionic
school, flourished (Enopides of Chios. Proclus ascribes to him
the solution of the following problems : From a point without,
to draw a perpendicular to a given line, and to draw an angle
on. a line equal to a given augle. That a man could gain a
reputation by solving problems so elementary as these, indi-
eates that geometry was still in its infancy, and that the
Greeks had not yet gotten far beyond the Egyptian con
structions.
The Ionic school lasted over one hundred years. The
pt ogress of mathematics during that period was slow, as
compared with its growth in a later epoch of Greek history.
A new impetus to its progress was given by Pythagoras.
TJie School of Pythagoras.
Pyrthagoras (580 ?-500? B.C.) was one of those figures which
impressed the imagination of succeeding /ffmes to such an
eitenlt that their real histories have become difficult to be
,d&a| med through the mythical haze that envelops them. The
jtello^dng account of Pythagoras excludes the most doubtful
20 A HISTORY OF MATHEMATICS.
statements. He was a native of Samos, and was drawn by
the fame of Pherecydes to the island of Syros. He then
visited the ancient Thales, who incited him to stndy in Egypt.
He sojourned in Egypt many years, and may have, visited
Babylon. On his return to Samos, he found it under the
tyranny of Polycrates. Failing in an attempt to found a
school there, he quitted home again and, following the current
of civilisation, removed to Magna Grsecia in South Italy. He
settled at Croton, and founded the famous Pythagorean school.
This was not merely an academy for the teaching of philosophy,
mathematics, and natural science, but it was a brotherhood,
the members of which were united for life. This brotherhood
l|ad observances "approaching masonic peculiarity. Thejr wore
forbidden to divulge the discoveries and doctrines of their
school. Hence we are obliged to speak of the Pythagoreans
as a body, and find it difficult to determine to whom each
particular discovery is to be ascribed. The Pythagoreans
themselves were in the habit of referring every discovep" back
to the great founder of the sect.
This school grew rapidly and gained considerable political
ascendency. But the mystic and secret obseivaaptcfe, intro
duced in imitation of Egyptian usages, and the a*|stooratic
tendencies of the school, caused it to becoiae ai* object, of
suspicion. The democratic party in Lower Itely revolted and
destroyed the buildings of the Pythagorean school*
ras fled to Tarentum and thence to Metapontum,
murdered.
Pythagoras has left behind no mathematical tventtees, and
our sources of information are rather scanty. Certain it is
that, in the Pythagorean school, mathematics was the
study. Pythagoras raised mathematics to the taak of a so
Arithmetic was courted by him as fervently ft&*geo$tetYj
fact, arithmetic is the foundation of his philosophic
icipal
iencc.
THE GrKEEKS. 21
The Eudemiart Summary says that "Pythagoras changed
the study of geometry into the form of a liberal education,
for he examined its principles to the bottom, and investigated
its theorems in an immaterial and intellectual manner." His
geometry was connected closely with his arithmetic. He was
especially fond of those geometrical relations which admitted
of arithmetical expression.
Like Egyptian geometry, the geometry of the Pythagoreans
. is much concerned with areas. To Pythagoras is ascribed the
important theorem that the square on the hypotenuse of a
right triangle is equal to the sum of the squares on the other
, two sides/ He had probably learned from the Egyptians the
truth of the theorem in the special case when the sides are
3, 4, 6, respectively. The story goes, that Pythagoras was so
jubilant over this discovery that he sacrificed a hecatomb. Its
authenticity is doubted, because the Pythagoreans believed in
the transmigration of the soul and opposed, therefore, the
shedding of blood. In the later traditions of the !N"eo-Pythago-"
reans this objection is removed by replacing this bloody sacri
fice by that of an ox made of flour " ! The proof of the law
,of three squares, given in Euclid s Elements, I. 47, is due to
Euclid himself, and not to the Pythagoreans. What the Py
thagorean method of proof was has been a favourite topic for
conjecture.
The theorem on the sum of the three angles of a triangle,
presumably known to Thales, was proved bythe Pythagoreans
after the manner of Euclid. They demonstrated also that the
plane about a point is completely filled by six equilateral
triangles, four squares, or three regular hexagons, so that it
is possible to divide up a plane into figures of either kind.
From the equilateral triangle and the square arise the solids,
namely the tetraedron, octaedron, icosaedron, and the cube.
These solids were, in all probability, known to the Egyptians,
A HISTORY OF MATHEMATICS.
excepting, perhaps, the icosaedron. In Pythagorean philos
ophy, they represent respectively the four elements of the
physical world; namely, fire, air, water, and earth. Later
another regular solid was/ discovered, namely the dodecaedron,
which, in absence of a/fifth element, was made to represent
the universe itself. lamblichus states that Hippasus, a Py-
thagorean, perished in the sea, "because he boasted that he first
divulged " the sphere with the twelve pentagons." The star-
f shaped pentagram was used as a symbol of recognition by the
1 Pythagoreans, and was called by them Health.
Pythagoras called the sphere the most beautiful of all solids,
and the circle the most beautifttl of all plane figures. The
treatment of the subjects of proportion and of irrational
quantities by him and his school will be taken up under the
head of arithmetic.
According to Eudemus, the Pythagoreans invented the prob-*
lerns concerning the application of areas, including the cases
~f defect and excess, as in Euclid, VI. 28, 29.
They were also familiar with the construction of a polygon
iqual in area to a given polygon and similar to another given
)olygon. This problem depends upon several important and
somewhat advanced theorems, and testifies to the fact that t
jhe Pythagoreans made no mean progress in geometry.
Of the theorems generally ascribed to the Italian school,
some cannot be attributed to Pythagoras himself, no* to his
earliest successors. The progress from empirical to reasoned
solutions must, of necessity, have been slow. It is worth
noticing that on the circle no theorem of any importance *wa$
discovered by this school, ,
Though politics broke up the Pythagorean fraternity, yet
the school continued to exist at least two centuries longer*
Among the later Pythagoreans, Philolaus and Arckytas aw
the most prominent. Philolaus wrote a book on the Pythago*
THE GREEKS. 23
rean doctrines. By him were first given to tlie world tlie
teachings of the Italian school, which had been kept secret
for a whole century. The brilliant Archytas of Tarentum
(428-347 B.C.), known as a great statesman and general, and
universally admired for his virtues, was the only great geome
ter among the Greeks when Plato opened his school. Archy-
tas was the first to apply geometry to mechanics and to treat
the latter subject methodically. He also found a very ingeni
ous mechanical solution to the problem of the duplication of
the cube. His solution involves clear notions on the genera
tion of cones and cylinders. This problem reduces itself to
finding two mean proportionals between two given lines.
These mean proportionals were obtained by Archytas from
the section of a half-cylinder. The doctrine of proportion
was advanced through him.
There is every reason to believe that the later Pythagoreans
exercised a strong influence on the study and development of
mathematics at Athens. The Sophists acquired geometry from
Pythagorean sources. Plato bought the works of Philolaus,
and had a warm friend in Archytas.
The Sophist School
After the defeat of the Persians under Xerxes at^the battle
of Salamis, 480 B.C., a league was formed among the Greeks
fco preserve the freedom of the now liberated Greek cities on
bhe islands and coast of the JEgsean Sea. Of this league
Athens soon became leader and dictator. She caused the
separate treasury of the league to be merged into that of
Athens, and then spent the money of her allies for her own
tggrandisement. Athens was also a great commercial centre.
Phus she became the richest and most beautiful city of an-
iquity. All menial work was performed by slaves. The
24 * A HISTORY OF MATHEMATICS.
citizen of Athens was well-to-do and enjoyed a large amount
of leisure. The government being purely democratic, every
citizen was a politician. To make his influence felt among
his fellow-men he must, first of all, be educated. Thus there
arose a demand for teachers. The supply came principally
from Sicily, where Pythagorean doctrines had spread. These
teachers were called Sophists, or "wise men." Unlike the
Pythagoreans, they accepted pay for their teaching. Although
rhetoric was the principal feature of their instruction, they
also taught geometry, astronomy, and philosophy. Athens
soon became the headquarters of Grecian men of letters, and
of mathematicians in particular. The home of mathematics
among the. Greeks was first in the Ionian Islands, then in
Lower Italy, and during the time now under consideration,
at Athens,
\ The geometry of the circle, which had been entirely
neglected by the Pythagoreans, was taken up by the Sophists.
Nearly all their discoveries were made in connection with
their innumerable attempts to solve the following three
famous problems :
(1) To trisect an arc or an angle.
(2) To " double the cube," i.e. to find a cube whose volume
is double that of a given cube.
(3) To "square the circle," i.e. to find a square or some
other rectilinear figure exactly equal in area to a given circle*
These problems have probably been the subject of more
discussion and research than any other problems m mathe
matics. The bisection of an angle was one of the easiest
problems in geometry. The trisection of an angle, on the
other hand, presented unexpected difficulties. A right iwagle
had been divided into three equal parts by the Pythagoreans,
But the general problem, though easy in appearanee^ tran
scended the power, of elementary geometry. Among the firfit
THE GREEKS. 25
fco wrestle with it was Hippias of Blis, a contemporary of
Socrates, and born about 460 B.C. Like all the later geome
ters, he failed in effecting the trisection by means of a ruler
and compass only. Prockts mentions a man, Hippias, presum
ably Hippias of Elis, as the inventor of a transcendental curve
which served to divide an angle not only into three, but into
any number of equal parts. This same curve was used later
by Deinostratus and others for the quadrature of the circle.
On this account it is called the quadratrix.
The Pythagoreans had shown that the diagonal of a square
is the side of another square having double the area of the
original one. This probably suggested the problem of the
duplication of the cube, i.e. to find the edge of a cube having
double the volume of a given cube. Eratosthenes ascribes to
this problem a different origin. The Delians were once suf
fering from a pestilence and were ordered by the oracle to
double a certain cubical altar. Thoughtless workmen simply
constructed a cube with edges twice as long, but this did not
pacify the gods. The error being discovered, Plato was con
sulted on the matter. He and his disciples searched eagerly
for a solution to this "Delian Problem." Hippocrates of Chios
(about 430 B.C.), a talented mathematician, but otherwise slow
and stupid, was the first to show that the problem could be
reduced to finding two mean proportionals between a given
line and another twice as long. For, in the proportion a: a?
= x : y = y : 2 a, since a? 2 = ay and y 2 = 2 ax and ce* = a 2 /, we
have a; 4 = 2 cfx and a? 3 = 2 a 8 . But he failed to find the two
mean proportionals. His attempt to square the pircl& was
also a failure; for though lie made himself celebrated by
squaring a kine, he committed an error in attempting to apply
this result to the squaring of the circle.
lujhis study of the quadrature and duplication-problems,
contributed much to the geometry of the circle.
26 A HISTORY OF MATHEMATICS.
The subject of similar figures was studied and partly
developed by Hippocrates. This involved the theory of
proportion. Proportion had, thus far, been used by the
Greeks only in numbers. They never succeeded in uniting
the notions of numbers and magnitudes. The term "number "
was used by them in a restricted sense. What we call
irrational numbers was not included under this notion. Not
even rational fractions were called numbers. They used the
word in the same sense as wo use "integers." Hence num
bers were conceived as discontinuous, while magnitudes were
continuous. The two notions appeared; therefore, entirely
distinct. The chasm between them is exposed to full view
in the statement of Euclid that "incommensurable magni
tudes do not have the same ratio as numbers." In Euclid s
Elements we find the theory of proportion of magnitudes
developed and treated independent of that of numbers. The
transfer of the theory of proportion from numbers to mag
nitudes (and to lengths in particular) was a difficult and
important step.
Hippocrates added to his fame by writing a geometrical
text-book, called the Elements. This publication shows that
the Pythagorean habit of secrecy was being abandoned;
secrecy was contrary to the spirit of Athenian life.
The Sophist Antiphon, a contemporary of Hippocrates, intro
duced the process of exhaustion for the purpose of solving
the problem of the quadrature. Ho did himself credit by
remarking that by inscribing in a circle a square, and oa its
sides erecting isosceles triangles with their vertices itt the
circumference, and on the sides of these triangles erecting
new triangles, etc., one could obtain a succession of .regular
polygons of 8, 16, 32, 64 sides, and so on, of "which eneh,
approaches nearer to the circle than the pxeviot^. o&f until
the circle is finally exhausted. Thais is obtained an iTD0 ^e
THE GREEKS* 27
polygon whose sides coincide with the circumference. Since
there can be found squares equal in area to any polygon,
there also can be found a square equal to the last polygon
inscribed, and therefore equal to the circle itself. Brys0n
of Heraclea, a contemporary of Antiphon, advanced the prob
lem of the quadrature considerably by circumscribing poly
gons at the same time that he inscribed polygons.- He erred,
however, in assuming that the area of a circle was the arith
metical mean between circumscribed and inscribed polygons.
Unlike Bryson and the rest of Greek geometers, Antiphon
seems to have believed it possible, by continually doubling
the sides of an inscribed polygon, to obtain a polygon coin
ciding with the circle. This question gave rise to lively
disputes in Athens. If a polygon can coincide with the
circle, then, says Simplicius, we must put aside the notion
that magnitudes are divisible ad infinitum. Aristotle always
supported the theory of tihe infinite divisibility, while Zeno,
the Stoic, attempted to show its absurdity by proving that
if magnitudes are infinitely divisible, motion is impossible.
Zeno argues that Achilles could not overtake a tortoise; for
while he hastened to the place where the tortoise had been
when he started, the tortoise crept some distance ahead, and
while Achilles reached that second spot, the tortoise again
moved forward a little, and so on. Thus the tortoise was
always in advance of Achilles. Such arguments greatly con
founded Greek geometers. No wonder they were deterred
by such paradoxes from introducing the idea of infinity into
their geometry. It did not suit the rigour of their proofs.
The process of Antiphon and Bryson gave rise to the cum
brous but perfectly rigorous "method of exhaustion." In
determining the ratio of the areas between two curvilinear
plane i|jp,|% s&y/two circles, geometers first inscribed or
Similar t>olverons, and then bv infyrAfl.ainar i
A HISTOEY OF MATHEMATICS.
the number of sides, nearly exhausted the spaces
between the polygons and circumferences. IProm the theo
rem that similar polygons inscribed in circles are to each
othsr as the squares on their diameters, geometers may have
divined the theorem attributed to Hippocrates of Chios that
the circles, which differ but little from the last drawn poly
gons, must be to each other as the squares on their diameters.
But in order to exclude all vagueness and possibility of doubt,
later Greek geometers applied reasoning like that in Euclid,
XII. 2, as follows : Let and c, D and d be respectively the
circles and diameters in question. Then if the proportion
D 2 : d 2 = C : c is not true, suppose that D 2 : $ = : c . If d < c,
then a polygon p can be inscribed in the circle c which conies
nearer to it in area than does c f . If P be the corresponding
polygon in C, then P : p = D 2 ; d 2 = G : c , and P : O = p : c .
Since j> > c f , we have P>C, which is absurd. Next they
proved by this same method of reductio ad absurdum the
falsity of the supposition, that c f > c. Since c can be neither
larger nor smaller than, c, it must be equal to it, QJE.D.
Hankel refers this Method of Exhaustion back to Hippo
crates of Chios, but the reasons for assigning it to this early
writer, rather than to Eudoxus, seem insufficient.
Though progress in geometry at this period is traceable only
at Athens, yet Ionia, Sicily, Abdera in Thrace, and Gyrene
produced mathematicians who made creditable contribution B
to the science. We can mention here only Bemociitus of
Abdera (about 460-370 B.C.), a pupil of Anaxagoras, a friend
of Philolaus,- and an admirer of the Pythagoreans. He
visited Egypt and perhaps even Persia. Ho was a successful
geometer and wrote on incommensurable lines, on geometry,
on numbers, and on perspective. Hone of these works are
extant, He used to boast that in the construction of plane
figures with proof no one had yet surpassed him, not even
THE GREEKS. 29*
the so-called harpedonaptae (" rope-stretchers ") of Egypt. By
this assertion he pays a flattering compliment to the skill
and ability of the Egyptians.
TJie Platonic School.
During the Peloponnesian War (431-404 B.C.) the progress
of geometry was checked. After the war, Athens sank into
the background as a minor political power, but advanced more
and more to the front as the leader in philosophy, literature,
and science. Plato was born at Athens in 429 B.C., the year
of the great plague, and died in 348. He was a pupil and
near friend of Socrates, but it was not from him that he
acquired his taste for mathematics. After the death of Soc
rates, Plato travelled extensively. In Cyrene he studied
mathematics under Theodoras. He went to Egypt, then to
Lower Italy and Sicily, where he came in contact with the
Pythagoreans. Archytas of Tarentum and Timaeus of Locri
became his intimate friends. On his return to Athens^ about
389 B.C., he founded his school in the groves of the Academia,
and devoted the remainder of his life to teaching and writing.
Plato s physical philosophy is partly based on that of the
Pythagoreans. Like them, he sought in arithmetic and
geometry the key to the universe. When questioned about
the occupation of the Deity, Plato answered that " He geom-
etrises continually." Accordingly, a knowledge of geometry
is a necessary preparation for the study of philosophy. To
show how great a value he put on mathematics and how
necessary it is for higher speculation, Plato placed the inscrip
tion over Ms porch, "Let no one who is unacquainted with
geometry enter here," Xenocrates, a successor of Plato as
teacher in the Academy, followed in his master s footsteps, by
declining to admit a pupil who had no mathematical training,
30 A HISTOBY OF MATHEMATICS.
with the remark, "Depart, for thou hast not the grip of
philosophy. 1 Plato observed that geometry trained the mind
for correct and vigorous thinking. Hence it was that the
Eudemian Summary says, " He filled his writings with mathe
matical discoveries, and exhibited on every occasion the re
markable connection between mathematics and philosophy."
With Plato as the head-master, we need not wonder that
the Platonic school produced so large a number of mathemati
cians. Plato did little real original work, but he made
valuable improvements in the logic and methods employed
in geometry. It is true that the Sophist geometers of the
previous century were rigorous in their proofs, but as a rule
they did not reflect on the inward nature of their methods.
They used the axioms without giving them explicit expression,
and the geometrical concepts, such as the point, line, surface,
etc., without assigning to them formal definitions, The Py
thagoreans called a point "unity in position/ 7 but this is a
statement of a philosophical theory rather than a definition.
Plato objected to calling a point a " geometrical fiction." He
defined a point as the "beginning of a line" or as "an indivis
ible line," and a line as " length without breadth." He called
the point, line, surface, the boundaries of the line, surface,
solid, respectively. Many of the definitions in Euclid are to
be ascribed to the Platonic school. The same is probably
true of Euclid s axioms. Aristotle refers to Plato the axiom
that "equals subtracted from equals leave equals."
7 One of the greatest achievements of Plato and his school is
the invention of analysis as a method of proof. To be sure,
this method had been used unconsciously by Hippocrates and
others ; but Plato, like a true philosopher^ turned the instinc
tive logic into a conscious, legitimate method.
The terms synthesis and analysis are used in mathematics
in a more special sense than in logic. In ancient mathematics
THE GREEKS. 31
they had a different meaning from what they now have. The
oldest definition of mathematical analysis as opposed to syn
thesis is that given in Euclid, XIII. 5, which in all probability
was framed by Eudoxus : " Analysis is the obtaining of the
thing sought by assuming it and so reasoning up to an
admitted truth ; synthesis is the obtaining of the thing
sought by reasoning up to the inference and proof of it."
The analytic method is not conclusive, unless all operations
involved in it are known to be reversible. To remove all
doubt, the Greeks, as a rule, added to the analytic process
a synthetic one, consisting of a reversion of all operations
occurring in the analysis. Thus the aim of analysis was to
aid in the discovery of synthetic proofs or solutions.
; Plato is said to have solved the problem of the duplication
of the cube. But the solution is open to the very same objec
tion which he made to the solutions by Archytas, Eudoxus,
and Menaeclmius. He called their solutions not geometrical,
but mechanical, for they required the use of other instruments
than the ruler and compass. He said that thereby " the good
of geometry is set aside and destroyed, for we again reduce it
to the world of sense, instead of elevating and imbuing it with
the eternal and incorporeal images of thought, even as it is
employed by God, for which reason He always is God." These
objections indicate either that the solution is wrongly attrib
uted to Plato or that he wished to show how easily non-geo
metric solutions of that character can be found. It is now
generally admitted that the duplication problem, as well as
the trisection and quadrature problems, cannot be solved by
means of the ruler and compass only.
Plato gave a healthful stimulus to the study of stereometry,
which until his time had been entirely neglected. The sphere
and the regular solids had been studied to some extent, but
the prism, pyramid, cylinder, and cone were hardly known to
32 A HISTOBY OF MATHEMATICS.
exist. All these solids became the subjects of investigation
by the Platonic school. One result of these inquiries was
epoch-making. Menaechmus, an associate of Plato and pupil
of Eudoxus, invented the conic sections, which, in course of
only a century, raised geometry to the loftiest height which
it was destined to reach during antiquity. Mensechmus cut
three kinds of cones, the right-angled/ acute-angled/ and
obtuse-angled/ by planes at right angles to a side of the
cones, and thus obtained the three sections which we now call
the parabola, ellipse, and hyperbola. Judging from the two
very elegant solutions of the "Delian. Problem" by means of
intersections of these curves, Mensechimis must have succeeded
well in investigating their properties.
Another great geometer was Dinostratus, the brother of
Menaechmus and pupil of Plato. Celebrated is his mechanical
solution of the quadrature of the circle, by means of the quad-
ratri of Hippias.
Perhaps the most brilliant mathematician of this period was
Eudoxus. He was born at Cniclus about 408 B.O., studied under
Archytas, and later, for two months, under Plato. He was
imbued with a true spirit of scientific inquiry, and has beea
called the father of scientific astronomical observation. From
the fragmentary notices of his astronomical researches, found
in later writers, Ideler and Schiaparolli succeeded in recon
structing the system of Eudoxus with its celebrated representa
tion of planetary motions by "concentric spheres*" Eudoxus
had a school at Cyzicus, went with his pupils to Athens, visit
ing Plato, and then returned to Cyzicxis, where ho died 355
B.C. The fame of the academy of Plato is to a large extent
due to Eudoxtts s pupils of the school at Cyzicua, aiaong
whom are Meneeclnnus, Dinostratus, Athensaus, and Helicon.
Diogenes Laertius describes Eudoxus as astronomer, physician,
legislator, as well as geometer. The Eudemimi Summary
THE GREEKS. 33
says that Eudoxus " first increased the number of general
theorems, added to the three proportions three more, aixd
raised to a considerable quantity the learning, begun by Plato,
on the subject of the section, to which he applied the analyt
ical method." By this c section is meant, no doubt, the
"golden section" (sectio aurea), which cuts a line in extreme
and mean ratio. The first five propositions in Euclid XIII.
relate to lines cut by this section, and are generally attributed
to Eudoxus. Eudoxus added much to the knowledge of solid
geometry. He proved, says Archimedes, that a pyramid is
exactly one-third of a prism, and a cone one-third of a cylinder,
having equal base and altitude. The proof that spheres are
to each other as the cubes of their radii is probably due to
him. He made frequent and skilful use of the method of
exhaustion, of which he was in all probability the inventor.
A scholiast on Euclid, thought to be Proclus, says further that
Eudoxus practically invented the whole of Euclid s fifth book.
Eudoxus also found two mean proportionals between two
given lines, but the method of solution is not known.
Plato has been called a maker of mathematicians. Besides
the pupils already named, the Eudemian Summary men
tions the following: Theaetetus of Athens, a man of great
natural gifts, to whom, no\loubt, Euclid was greatly indebted
in the composition of the 10th book ; 8 treating of incommensu-
rables ; Leodamas of Thasos ; Feocleides and his pupil Leon,
who added much to the work of their predecessors, for Leon
wrote an Elements carefully designed, both in number and
utility of its proofs; Theudius of Magnesia, who composed a
very good book of Elements and generalised propositions,
which had been confined to particular cases ; Hermotimus of
Colophon, who discovered many propositions of the Elements
and composed some on loci; and, finally, the names of Amyclas
of Heraclea, Cyzicenus of Athens, and Philippus of Mende.
34 A HISTOBY OF MATHEMATICS,
A skilful mathematician of whose life and works we have
no details is Aristaelis, the elder, probably a senior contempo
rary of Euclid. The fact that he wrote a work on conic
sections tends to show that much progress had been made in
their study during the time of Menaechmus. Aristous wrote
also on regular solids and cultivated the analytic method.
His works contained probably a summary of the researches
of the Platonic school. 8
Aristotle (384-322 B.C.), the systematise! of deductive logic,
though not a professed mathematician, promoted the science
of geometry by improving some of the most difficult defini
tions. His Physics contains passages with suggestive hints
of the principle of virtual velocities. About his time there
appeared a work called Mechanic, of which he is regarded
by some as the author. Mechanics was totally neglected by
the Platonic school.
The First Alexandrian School,
In the previous pages we have seen the birth of geometry
in Egypt, its transference to the Ionian Islands, thence to
Lower Italy and to Athens. Wo have witnessed its growth
in Greece from feeble childhood to vigorous manhood, and
now we shall see it return to the land of its birth and there
derive new vigour.
During her declining years, immediately following the
Feloponnesian War, Athens produced the greatest scientists
and philosophers of antiquity. It was the timo of Plato
and Aristotle. In 338 B.C., at the battle of OUf&ronea, Athens
was beaten, by Philip of Macedon, and her power was broken
forever. Soon after, Alexander the Great, the son of Philip,
started out to conquer the world. la eleven years he built
up a great empire which broke to pieep ia a day*
THE GREEKS. 35
fell to the lot of Ptolemy Soter. Alexander had founded
the seaport of Alexandria, which soon became "the noblest
of all cities." Ptolemy made Alexandria the capital. The
history of Egypt during the next three centuries is mainly
the history of Alexandria. Literature, philosophy, and art
were diligently cultivated. Ptolemy created the university
of Alexandria. He founded the great Library and built labo
ratories, museums, a zoological garden, and promenades. Alex
andria soon became the great centre of learning.
Demetrius Phalereus was invited from Athens to take
charge of the Library, and it is probable, says Gow, that
Euclid was invited with him to open the mathematical school.
Euclid s greatest activity was during the time of the first
Ptolemy, who- reigned from 306 to 283 B.C. Of the life of
Euclid, little is known, except what is added by Proclus to
the Eudemian Summary. Euclid, says Proclus, was younger
than Plato and older than .Eratosthenes and Archimedes, the
latter of whom mentions him. He was of the Platonic sect, and
well read in its doctrines. He collected the Elements, put
in order much that Eudoxus had prepared, completed many
things of Theaetetus, and was the first who reduced to unob
jectionable demonstration, the imperfect attempts of his prede
cessors. When Ptolemy once asked him if geometry could
not be mastered by an easier process than by studying the
Elements, Euclid returned the answer, "There is no royal
road to geometry." Pappus states that Euclid was distin
guished by the fairness and kindness of his disposition, par
ticularly toward those who could do anything to advance
the mathematical sciences. Pappus is evidently making a
contrast to Apollonius, of whom he more than insinuates the
opposite character. 9 A pretty little story is related by Sto-
baeus: 6 "A youth who had begun to read geometry with
Euclid, when h had learnt the first proposition, inquired,
36 A HISTORY OF MATHEMATICS.
< What do I get by learning these tilings ? So Euclid called
his slave and said, Give him threepence, since he must
make gain out of what he learns/ " These are about all the
personal details preserved by Greek writers, Syrian and
Arabian writers claim to know much more, but they are unre
liable. At one time Euclid of Alexandria was universally
confounded with Euclid of Megara, who lived a century
earlier.
The fame of Euclid has at all times rested mainly upon his
book on geometry, called the Elements. This book was so far
superior to the Elements written by Hippocrates, Loon, and
Theudius, that the latter works soon perished in the straggle
for existence. The Greeks gave Euclid the special title of
~ c the author of the .Elements" It is a remarkable fact in thei
xistory of geometry, that the Elements of Euclid, written two
thousand years ago, are still regarded by many as the best
ntroduction to the mathematical sciences. In England they
xre used at the present time extensively as a text-book in
schools. Some editors of Euclid have, however, been inclined
bo credit him with more than is his due. They would have
us believe that a finished and unassailable system of geometry
sprang at once from the brain of Euclid, " an armed Minerva
from the head of Jupiter." They fail to mention the earlier
eminent mathematicians from whom Euclid got his material.
Comparatively few of the propositions and proofs in the
Elements are his own discoveries. In fact, the proof of tlie
" Theorem of Pythagoras " is the only one directly ascribed to
him. Allman conjectures that the substance of Books I,, II,
IV. comes from the Pythagoreans, that tlie substance of Book
VI. is due to the Pythagoreans and Exidoatufy tlto latter con
tributing the doctrine of proportion as applicable to ineom-
mensurables and also the Method of Exhaustions (Book VII.),
that Thesetetus contributed much toward Books X, and XIII,
THE GREEKS. 37
that the principal part of the original work of Euclid himself
is to be found in Book X. 8 Euclid was the greatest systema-
tiser of his time. By careful selection from the material
before him, and by logical arrangement of the propositions
selected, he built up, from a few definitions and axioms, a
proud and lofty structure. It would be erroneous to believe
that he incorporated into his Elements all the elementary
theorems known at. his time. Archimedes, Apollonius, and
even he himself refer to theorems not included in his Ele
ments, as being well-known truths.
The text of the Elements now commonly used is Theon s
edition. Theon of Alexandria, the father of Hypatia, brought
out an edition, about 700 years after Euclid, with some altera
tions in the text. As a consequence, later commentators,
especially Robert Simson, who laboured under the idea that
Euclid must be absolutely perfect, made Theon the scape
goat for all the defects which they thought they could discover
in the text as they knew it. But among the manuscripts sent
by Napoleon I. from the Vatican to Paris was found a copy of
the Elements believed to be anterior to Theon s recension.
Many variations from Theon s version were noticed therein,
but they were not at all important, and showed that Theon
generally made only verbal changes. The defects in the
Elements for which Theon was blamed must, therefore, be
due to Euclid himself. The Elements has been considered as
offering models of scrupulously rigoroxis demonstrations. It
is certainly true that in point of rigour it compares favourably
with its modern rivals ; but when examined in the light of
strict mathematical logic, it has been pronounced by C. S.
Peirce to be " riddled with fallacies." The results are correct
only because the writer s experience keeps him on his guard.
At the beginning of our editions of the Elements, under
the head of definitions, are given the assumptions of such
38 A HISTORY OF MATHEMATICS.
notions as the point, line, etc., and some verbal explanations.
Then follow three postulates or demands, and twelve axioms.
The term axiom 7 was used by Proclus, but not by Euclid.
He speaks, instead, of common notions common either
to all men or to all sciences. There has been much contro
versy among ancient and modern critics on the postulates and
axioms. An immense preponderance of manuscripts and the
testimony of Proclus place the axioms ? about right angles
and parallels (Axioms 11 and 12) among tho postulates. 9 10
This is indeed their proper place, for they arc really assump
tions, and not common notions or axioms. Tho postulate
about parallels plays an important role in the history of non*
Euclidean geometry. The only postulate which Kxiolid missed
was the one of superposition, according to which figures
can, be moved about in space without any alteration in form
or magnitude.
The Moments contains thirteen books by Euclid, and two,
of which it is stipposed that Hypsicles and Damasoms are
the authors. The first four books are on plane geometry.
The fifth book treats of the theory of proportion as applied
to magnitudes in general. The sixth book develops the
geometry of similar figures. The seventh, eighth, ninth
booksy^re on the theory of numbers, or on arithmetic. In the
ninth book is found the proof to the theorem that tha number
of primes is infinite. The tenth book treats of the theory of
incommensurables. The next three books are on stereometry.
The eleventh contains its more elementary theorems ; the
twelfth, the metrical relations of the pyramid, prism, cone,
cylinder, and sphere. Tho thirteenth treats of the regular
polygons, especially of the triangle and pentagon, and then uses
them as faces of the five regular solids ; namely, the totraedron,
octaedron, icosaedron, cube, and dodecaedron. The regular
solids were studied so extensively by the.Platonists fhfrjfe they
THE GEBEKS. 39
received the name of "Platonic figures." The statement of
Proclns that the whole aim of Euclid in writing the Elements
was to arrive at the construction of the regular solids, is
obviously wrong. The fourteenth and fifteenth books, treat
ing of solid geometry, are apocryphal.
A remarkable feature of Euclid s, and of all Greek geometry
before Archimedes is that it eschews mensuration. Thus the
theorem that the area of a triangle equals half the product
of its base and its altitude is foreign to Euclid.
Another extant book of Euclid is the Data. It seems to
have been written for those who, having completed the Ele
ments, wish to acquire the power of solving new problems
proposed to them. The Data is a course of practice in analy
sis. It contains little or nothing that an intelligent student
could not pick up from the Elements itself. Hence it contrib
utes little to the stock of scientific knowledge. The following
are the other extant works generally attributed to Euclid:
Phenomena, a work on spherical geometry and astronomy;
Optics, which develops the hypothesis that light proceeds
from the eye, and not from the object seen; Catoptrica, con
taining propositions on reflections from mirrors ; De Divisioni-
ftus, a treatise on the division of plane figures into parts
having to one another a given ratio ; Sectio Canonis, a work
on musical intervals. His treatise on Porisms is lost ; but
much learning has been expended by Eobert Sims on and
M. Ohasles in restoring it from numerous notes found in the
writings of Pappus. The term porism is vague in meaningl
The aim of a porism is not to state some property or truth,
like a theorem, nor to effect a construction, like a problem,
but to find and bring to view a thing which necessarily exists
with given numbers or a given construction, as, to find the
centre of a given circle, or to find the G.C.D. of two given
numbers. 6 His other lost works are Fallacies, containing
40 A HISTORY OF MATHEMATICS.
exercises in detection of fallacies; Conic Sections, in four
books, which are the foundation of a work on the same sub
ject by Apollonius; and Loci on a Surface, the meaning of
which title is not understood. Heiberg believes it to mean
"loci which are surfaces."
The immediate successors of Euclid in the mathematical
school at Alexandria were probably Conon, Dositheus, and
Zeuxippus, but little is known of them.
\ Archimedes (287?~212 B.C.), the greatest mathematician of
antiquity, was born in Syracuse. Plutarch calls him a rela
tion of King Hieronj but more reliable is the statement of
Oicero, who tells us he was of low birth. Diodorus says he
visited Egypt, and, since he was a great friend of Conon and
Eratosthenes, it is highly probable that he studied in. Alexan
dria. This belief is strengthened by the fact that he had
bhe most thorough acquaintance with all the work previously
done in mathematics. He returned, however, to Syracuse,
where he made himself useful to his admiring friend
patron, King Hieron, by applying his extraordinary inventive
genius to the construction of various war-engines, by wjbdch
he inflicted much loss on the Romans during the siege of
Marcellus. 1 The story that, by the use of mirrors reflecting
bhe sun s rays, he set on fire the Roman ships, when they
came within bow-shot of the walls, is probably a fiction. tJTIxe
city was taken ait length "by the Romans, and Archimedes
perished in the indiscriminate slaughter which followed. Ac
cording to tradition, he was, at the time, studying the diagram
bo some problem drawn in the sand. As a Roman soldier
approached him, he called out, "Don t spoil my circles."
The soldier, feeling insulted, rushed upon him and killed
him. Jf No -blame attaches to [the Roman general Marcelltts,
who admired his genius, and raised in his honour a tomb
bearing the figure of a sphere inscribed in a cylinder. When
THE GREEKS. 41
Cicero was in Syracuse, lie found the tomb buried under
rubbish.
Archimedes was admired by his fellow-citizens chiefly for"""
his mechanical inventions ; he himself prized far more highly
his discoveries in pure science. He declared that "every kind
of art which was connected with daily needs was ignoble and
vulgar-^" Some of his works have been lost. The following
are the extant books, arranged approximately in chronological
order : 1. Two books on Equiponderance of Planes or Centres
of Plane Gravities, between which is inserted his treatise or.
the Quadrature of the Parabola; 2. Two books on the Sphere
and Cylinder; 3. The Measurement of the Circle ; 4. On Spirals;
5. Conoids and Spheroids; 6. The Sand-Counter; 7. Two books
on Floating Bodies; 8. Fifteen Lemmas, j
In the book on the Measurement of the Circle, Archimedes
proves first that the area of a circle is equal to that of a
right triangle having the length of the circumference for its
b~se, and the radius for its altitude. In this he assumes that
there exists a straight line equal in length to the circumference
an assumption objected to by some ancient critics, on
the ground that it is not evident that a straight line can equal
a curved one. The finding of suct^ a line was the next prob
lem. He fir^t finds an upper limit to the ratio of the circum
ference to the diameter, or TT. To do this, he starts with an
equilateral triangle of which the base is a tangent and the
vertex is the centre of the circle. By successively bisecting
the angle at the centre, by comparing ratios, and by taking the
irrational square roots always a little too small, he finally
arrived at the conclusion that ?r<3^. Next he finds a lower
limit by inscribing in the circle regular polygons of 6, 12, 24,
48, 96 sides, finding for each successive polygon its perimeter,
which is, of course, always less than the circumference. Thus
he finally concludes that "the circumference of a circle ex-
42 A HISTORY OF MATHEMATICS.
ceeds three times its diameter by a part which, is less than $
but more than f& of the diameter." This approximation is
exact enough for most purposes.
The Quadrature of the Parabola contains two solutions to
the problem one mechanical, the other geometrical. The
method of exhaustion is used in both.
Archimedes studied also the ellipse and accomplished its
quadrature, but to the hyperbola he seems to have paid less at
tention. It is believed that he wrote a book on conic sections.
J~-Of all his discoveries Archimedes prized most highly those
in his Sphere and Cylinder. In it are proved the new
theorems, that the surface of a sphere is equal to four times
a great circle ; that the surface of a segment of a sphere is
equal to a circle whose radius is the straight line drawn from
the vertex of the segment to the circumference of its basal
circle ; that the volume and the surface of a sphere are of,
the volume and surface, respectively, of the cylinder circum
scribed about the sphere. Archimedes desired that the figure
to the last proposition be inscribed on his tomb. This was
ordered done by Marcellus. }
<CThe spiral now called the "spiral of Archimedes," and
described in the book On Spirals, was discovered by Archi
medes, and not, as some believe, by his friend Conon. 8 His
treatise thereon is, perhaps, the most , wonderful of- all his
works. Nowadays, subjects of this kind are made easy by .
the use of the infinitesimal calculus. In its stead the aBteients,
used the method of exhaustion. Nowhere is the fertility of
his genius more grandly displayed than in his masterly use of
this method. With Euclid and his predecessors the method
of exhaustion was only the means of proving propositions
which must have been seen anf : b&litved - , before they were
proved. But in the hands of Arehtoete it lecame art instru
ment of discovery, 9
THE GKEEKS. 43
By the word conoid/ in his book on Conoids and
Spheroids, is meant thQ solid produced by the revolution
of a parabola or a hyperbola about its axis. Spheroids
are produced by the revolution of an ellipse, and are long
or flat, according as the ellipse revolves around the major
or minor axis. The book leads up to the cubature of these
solids. /
We Rave now reviewed briefly all his extant works on geom
etry. His arithmetical treatise and problems will be consid
ered later. We shall now notice his works on mechanics.
Archimedes is the author of the first sound knowledge on this
subject. Archytas, Aristotle, and others attempted to form
the known mechanical truths into a science, but failed. Aris
totle knew the property of the lever, but could -not establish
its true mathematical theory. The radical and fatal defect
in the speculations of the Greeks, says Whewell, was "that
though they had in their possession facts and ideas, the ideas
were not distinct and appropriate to the facts. 93 For instance,
Aristotle asserted that when a body at the end of a lever is
moving, it may be considered as having two motions ; one in
the direction of the tangent and one in the direction of the
radius ; the former motion is, he says, according to nature, the
latter contrary to nature. These inappropriate notions of
natural 5 and unnatural motions, together with the habits
of ^thought which dictated these speculations, made the per
ception, of the true grounds of mechanical properties impos
sible." It seems strange that even after Archimedes had
entered upon the right path, this science should have remained
absolutely stationary till the time of Galileo a period of
nearly two thousand years.
The proof of the property of the lever, given in his Equi-
ponderance of Planes, holds its place in text-books to this day.
His estimate of the efficiency of the lever is expressed in the
44 A HISTORY OF MATHEMATICS.
saying attributed to him, "Give me a fulcrum on which to
rest, and I will move the earth."
/ s "While the JSquiponderance treats of solids, or the equilib
rium of solids, the book 011 Floating Bodies treats of hydro
statics. His attention was first drawn to the subject of
specific gravity when King Hieron asked him to test whether
a crown, professed by the maker to be pure gold, was not
alloyed with silver."] The story goes that our philosopher was
in a bath when the true method of solution flashed on his
mind. He immediately ran home, naked, shouting, " I have
found it ! " llo solve the problem, he took a piece of gold and
a piece of silver, each weighing the same as the crown. Ac
cording to one author, he determined the volume of water
displaced by the gold, silver, and crown respectively, and
calculated from that the amount of gold and silver in the
crown. According to another writer, he weighed separately
the gold, silver, and crown, while immersed in water, thereby
determining their loss of weight in water. Prom these data
he easily found the solution. It is possible that Archimedes
solved the problem by both methods.
After examining the writings of Archimedes, one can well
understand how, in ancient times, an < Archimedean problem ?
came to mean a problem too deep for ordinary minds to solve,
and how an i Archimedean proof came to be the synonym for
unquestionable certainty. Archimedes wrote on a very wide
range of subjects, and displayed great profundity in each. He
is the Newton of antiquity/]
Eratosthenes, eleven years younger than Archimedes, was a
native of Cyrene. He was educated in, Alexandria under
Callimachus the poet, whom he succeeded as custodian of
the Alexandrian Library. His many-sided activity may be
inferred from his works. He wrote on Cfood and Evil, Meas
urement of the Earthy Comedy, Geography, Chronology, Constel-
THE GREEKS. 45
lotions, and the Duplication of the Cube. He was also a
philologian and a poet. He measured the obliquity of the
ecliptic and invented a device for finding prime numbers.
Of his geometrical writings we possess only a letter to
Ptolemy Euergetes, giving a history of the duplication prob
lem and also the description of a very ingenious mechanical
contrivance of his own to solve it. In his old age he lost
his eyesight, and on that account is said to have committed
suicide by voluntary starvation.
About forty years after Archimedes flourished Apollonius of
Perga, whose genius nearly equalled that of his great prede
cessor. He incontestably occupies the second place in dis
tinction among ancient mathematicians. Apollonius was
born in the reign of Ptolemy Euergetes and died under
Ptolemy Philopator, who leigned 222-205 B.C. He studied at
Alexandria under the successors of Euclid, and for some time/
also, at Perganmm, where he made the acquaintance of that
Eudemus to whom he dedicated the first three books of his
Conic Sections. The brilliancy of his great work brought him
the title of the " Great Geometer." This is all that is known
of his life.
His Conic Sections were in eight books, of which the first
four only have come down to us in the original Greek. The
next three books were unknown in Europe till the middle of
the seventeenth century, when an Arabic translation, made
about 1250, was discovered. The eighth book has never been
found. In 1710 Halley of Oxford published the Greek text
of the first four books and a Latin translation of the remain
ing three, together with his conjectural restoration of the
eighth book, founded on the introductory lemmas, of Pappus.
The first four books contain little more than the substance
of what earlier geometers had done. Eutocius tells us that
Heraclides, in his life of Archimedes, accused Apollonius of
46 A HISTOEY OF MATHEMATICS.
having appropriated, in his Conic Sections, the unpublished
discoveries of that great mathematician. It is difficult to
believe that this charge rests upon good foundation. Eutocius
quotes Geminus as replying that neither Archimedes nor
Apollonius claimed to have invented the conic sections, but
that Apollonius had introduced a real improvement. While
the first three or four books were founded on the works of
Menaechmus, Aristseus, Euclid, and Archimedes, the remaining
ones consisted almost entirely of new matter. The first three
books were sent to Eudemus at intervals, the other books
(after Eudemus s death) to one Attalus. The preface of the
second book is interesting as showing the mode in which
Greek books were ( published ? at this time. It reads thus :
" I have sent my son Apollonius to bring you (Eudemus) the
second book of my Conies. Bead it carefully and communi
cate it to such others as are worthy of it. If Philonides, the
geometer, whom I introduced to you at Ephesus, comes into
the neighbourhood of Pergamum, give it to him also." 12
The first book, says Apollonius in his preface to it, " con
tains the mode of producing the three sections and the conju
gate hyperbolas and their principal characteristics, more fully
and generally worked out than in the writings of other
authors." We remember that Mensechmus, and all his suc
cessors down to Apollonius, considered only sections of right
cones by a plane perpendicular to their sides, and that the
three sections were obtained each from a different cone.
Apollonius introduced an important generalisation. He pro
duced all the sections from one and the same cone, whether
right or scalene, and by sections which may or may not be
perpendicular to its sides. The old names for the three curves
were now no longer applicable. Instead of calling the three
curves, sections of the^ acute-angled/ * right-angled/ and
obtuse-angled cone, he called them ellipse, parabola, and
THE GREEKS. 47
hyperbola, respectively. To be sure, we find the words < parab
ola and ellipse ? in the works of Archimedes, but they are
probably only interpolations. The word ellipse > was applied
because y 2 <px,p being the parameter; the word parabola
was introduced because y 2 =px, and the term < hyperbola
because y*>px.
The treatise of Apollonius rests on a unique property of
conic sections, which is derived directly from the nature of
the cone in which these sections are found. How this property
forms the key to the system of the ancients is told in a mas
terly way by M. Chasles. 13 "Conceive," says he, "an oblique
cone on a circular base; the straight line drawn from its
summit to the centre of the circle forming its base is called
the axis of the cone. The plane jpassing through the axis,
perpendicular to* its base; exit s" the cone along two lines and
determines in the circle a diameter ; the triangle having this
diameter for its base and the two lines, for its sides, is called
the triangle through the axis. In the formation of his conic
sections, Apollonius supposed the cutting plane to be perpen
dicular to the plane of the triangle through the axis. The
points in which this plane meets the two sides of this triangle
are the vertices of the curve ; and the straight line which joins
these two points is a diameter of it. Apollonius called this
diameter latus transversum. At one of the two vertices of the
curve erect a perpendicular (latus rectum) ;to the plane of the
triangle through the axis, of a certain length, to be determined
as we shall specify later, and from the extremity of this per
pendicular draw a straight line to the other vertex of the
curve ; now, through any point whatever of the diameter of
the curve, draw at right angles an ordinate : the square of this
ordinate, comprehended between the diameter and the curve,
will be equal to the rectangle constructed on the portion of
the ordinate comprised between the diameter and the straight
4:8 A HISTOBY OF MATHEMATICS.
line, and the part of the diameter comprised between the first
vertex and the foot of the ordinate. Such is the characteristic
property which Apollonius recognises in his conic sections and
which he uses for the purpose of inferring from it, by adroit
transformations and deductions, nearly all the rest. It plays,
as we shall see, in his hands, almost the same rdle as the
equation of the second degree with two variables (abscissa and
ordinate) in the system of analytic geometry of Descartes.
"It will be observed from this that the diameter of the
curve and the perpendicular erected at one of its extremities
suffice to construct the curvj|r These are the two elements
which the ancients used, with which to establish their theory
of conies. The perpendicular in question was called by them
latus erectum; the moderns changed this name first to that of
latus rectum, and afterwards to that of parameter."
The first book of the Conic Sections of Apollonius is almost
wholly devoted to the generation of the three principal conic
sections.
The second book treats mainly* of asymptotes, axes, and
diameters.
The third book treats of the equality or proportionality
of triangles, rectangles, or squares, of which the component
parts are determined by portions of transversals, chords,
asymptotes, or tangents, which are frequently subject to a
great number of conditions. It also touches the subject of
foci of the ellipse and hyperbola.
In the fourth book, Apollonius discusses the harmonic divis
ion of straight lines. He also examines a system of two
conies, and shows that they cannot cut each other in more
than four points. He investigates the various possible relative
positions of two conies, as, for instance, when, they have one
or two points of contact with each other.
The fifth book reveals better than any other the giant
THE GREEKS. 49
intellect of its author. Difficult questions of maxima and
minima, of which, few examples are found in earlier works, are
here treated most exhaustively. The subject investigated is,
to find the longest and shortest lines that can he drawn from
a given point to a conic. Here are also found the germs of
the subject ofevolutes and centres of osculation.
The sixth book is on the similarity of conies.
The seventh book is on conjugate diameters.
The eighth book, as restored by Halley, continues the sub
ject of conjugate diameters.
It is worthy of notice that Apollonius nowhere introduces
the notion of directrix for a conic, and that, though he inciden
tally discovered the focus of an ellipse and hyperbola, he did
not discover the focus of a parabola. 6 Conspicuous in his
geometry is also the absence of technical terms and symbols,
which renders the proofs long and cumbrous.
The discoveries of Archimedes and Apollonius, says M.
Chasles, 13 marked the most brilliant epoch of ancient geometry.
Two questions which have occupied geometers of all periods
may be regarded as having originated with them. The first
of these is the quadrature of curvilinear figures, which gave
birth to the infinitesimal calculus. The second is the theory
of conic sections, which was the prelude to the theory of
geometrical curves of all degrees, and to that portion of
geometry which considers only the forms and situations
of figures, and uses only the intersection of lines and surfaces
and the ratios of rectilineal distances. These two great
divisions of geometry may be designated by the names of
Geometry of Measurements and Geometry of Forms and Situa
tions, or, Geometry of Archimedes and of Apollonius.
Besides the Conic Sections, Pappus ascribes to Apollonius
the following works: On Contacts, Plane Loci, Inclinations,
Section of an Area, Determinate Section, and gives lemmas
50 A HISTOEY OF MATHEMATICS.
from which attempts have been made to restore the lost
originals. Two books on De Sectione Rationis have been
found in the Arabic. The book on Contacts, as restored by
Vieta, contains the so-called " Apollonian Problem " : Given
three circles, to find a fourth which shall touch the three.
Euclid, Archimedes, .and Apollonius brought geometry to
as- high a state of perfection as it perhaps could be brought
without first introducing some more general and more powerful
method than the old method of exhaustion. A briefer sym
bolism, a Cartesian geometry, an infinitesimal calculus, were
needed. The Greek mind was not adapted to the invention of
general methods. Instead of a climb to still loftier heights
we observe, therefore, on the part of later Greek geometers, a
descent, during which they paused here and there to look
around for details which had been passed by in the hasty
ascent. 3
Among the earliest successors of Apollonius was Mcomedes.
Nothing definite is known of him, except that he invented the
conchoid (" mussel-like"). He devised a little machine by
which the curve could be easily described. With aid of the
conchoid he duplicated the cube. The curve can also be used
for trisecting angles in a way much resembling that* in the
eighth lemma of Archimedes. Proclus ascribes this mode of
trisection to Nicomedes, but Pappus, on the other hand, claims
it as his own. The conchoid was used by Newton in con
structing curves of the third degree.
About the time of Mcomedes, flourished also Diodes, the
inventor of the cissoid ("ivy-like"). This curve he used for
finding two mean proportionals between two given straight
lines.
About the life of Perseus we know as little as about that of
Nicomedes and Diocles. He lived some time between 200 and
100 B.C. Prom Heron and Geminus we learn that he wjtote a
THE GREEKS. 51
work omthe spire, a sort of anchor-ring surface described by
Heron as being produced by the revolution of a circle around
one of its chords as an axis. The sections of this surface
yield peculiar curves called spiral sections, which, according to
G-eminus, were thought out by Perseus. These curves appear
to be the same as the Hippopede of Eudoxus.
, Probably somewhat later than Perseus lived Zenodorus. He
wrote an interesting treatise on a new subject; namely, iso-
perimetncal figures. Fourteen propositions are preserved by
Pappus and Theon. Here are a few of them : Of isoperimet-
rical, regular polygons, the one having the largest number of
angles has the greatest area; the circle has a greater area than
any regular polygon of equal periphery ; of all isoperimetrical
polygons of n sides, the regular is the greatest ; of all solids
having surfaces equal in area, the sphere has TfieT^eatest**
volume.
Hypsicles (between 200 and 100 B.C.) was supposed to be
the author of both the fourteenth and fifteenth books of
Euclid, but recent critics are of opinion that the fifteenth
book was written by an author who lived several centuries
after Christ. The fourteenth book contains seven elegant
theorems on regular solids. A treatise of Hypsicles on Risings
is of interest because it is the first Greek work giving the
division of the circumference into 360 degrees after the fash
ion of the Babylonians.
Hipparchus of Nicsea in Bithynia was the greatest astron
omer of antiquity. He established inductively the famous
theory of epicycles and eccentrics. As might be expected, he
was interested in mathematics, not per se, but only as an aid
to astronomical inquiry. No mathematical writings of his
are extant, but Theon of Alexandria informs us that Hippar-
chus originated the science of trigonometry, and that he calcu
lated a " table of chords " in twelve books. Such calculations
52 A HISTORY OF MATHEMATICS.
must have required a ready knowledge of arithm } Sal and
algebraical operations.
About 155 B.C. flourished Heron the Elder of Alexandria.
He was the pupil of Ctesibius, who was celebrated for his
ingenious mechanical inventions, such as the hydraulic organ,
the water-clock, and catapult. It is believed by some that
Heron was a son of Ctesibius. He exhibited talent of the
same order as did his master by the invention of the eolipile
and a curious mechanism known as "Heron s fountain."
Great uncertainty exists concerning his writings. Most au
thorities believe him to be the author of an important Treatise
on the Dioptra, of which there exist three manuscript copies,
quite dissimilar. But M. Marie u thinks that the Dioptra is
the work of Heron the Younger, who lived in the seventh or
eighth century after Christ, and that Geodesy, another book
supposed to be by Heron, is only a corrupt and defective copy
of the former work. Dioptra, contains the important formula
for finding the area of a triangle expressed in terms of its
sides ; its derivation is quite laborious and yet exceedingly
ingenious. " It seems to me difficult to believe," says Chasles,
"that so beautiful a theorem should be found in a work so
ancient as that of Heron the Elder, without that some Greek
geometer should have thought to cite it," Marie lays great
stress on this .silence of the ancient writers, and argues from
it that the true author must be Heron the Younger or some
writer much more recent than Heron the Elder, But no reli
able evidence has been found that there actually existed a
second mathematician by the name of Herory
"Dioptra," says Venturi, were instramejrfs which had great
resemblance to our modern theodolites. ( The book Dioptra is
a treatise on geodesy containing solutions, with aid of these
^instruments, of a large number of questions in geometry, such
as to find the distance between two points, of which one only
THE GEBEKS. 53
is accessible, or between two points which are visible but both
inaccessible ; from a given point to draw a perpendicular to a
line which cannot be approached; to find the difference of
level between two points ; to measure the area of a field with
out entering it.
Heron was a practical surveyor. This may account for the
fact that his writings bear so little resemblance to those of
the Greek authors, who considered it degrading the science
to apply geometry to surveying. The character of his geom
etry is not Grecian, but decidedly Egyptian. This fact is the
more surprising when we consider that Heron demonstrated
his familiarity with Euclid by writing a commentary on the
Elements. 21 Some of Heron s formulas point to an old Egyp
tian origin. Thus, besides the above exact formula for the
area of a triangle in terms of its sides, Heron gives the for
mula a * "i" a * x -, which bears a striking likeness to the for-
mula i 2 x -^_ 2 for finding the area of a quadrangle,
jU
found in the Edfu inscriptions. There are, moreover, points
of resemblance between Heron s writings and the ancient
Ahmes papyrus. Thus Ahmes used unit-fractions exclusively ;
Heron uses them ^oftener than other fractions. Like Ahmes
and the priests at Edfu, Heron divides complicated figures
into simpler ones by drawing auxiliary lines; like them,
he shows, throughout, a special fondness for the isosceles
trapezoid.
The writings of Heron satisfied a practical wan^ and for
that reason were borrowed extensively by other peoples. We
find traces of them in Rome, in the Occident during the Middle
Ages, and even in India.
Geminus of Khodes (about 70 B.C.) published an astronomi
cal work still extant. He wrote also a book, now lost, on the
Arrangement of Mathematics, which contained many valuable
54 . A HISTORY OF MATHEMATICS.
ff
notices of the early history of Greek mathematics. Froclus
and Eutocius quote it frequently. Theodosius of Tripolis is
the author of a book of little merit on the geometry of the
sphere. Dionysodorus of Amisus in Pontus applied the inter
section of a parabola and hyperbola to the solution of a prob
lem which Archimedes, in his Sphere and Cylinder, had left
incomplete. The problem is "to cut a sphere so that its seg
ments shall be in a given ratio."
We have now sketched the progress of geometry down to
the time of Christ. Unfortunately, very little is known of
the history of geometry between the time of Apollonius and
the beginning of the Christian era. The names of quite a
number of geometers have been mentioned, but very few of
their works are now extant. It is certain, however, that there
were no mathematicians of real genius from Apollonius to
Ptolemy, excepting Hipparchus and perhaps Heron.
The Second Alexandrian School.
The close of the dynasty of the Lagides which ruled Egypt
from the time of Ptolemy Soter, the builder of Alexandria,
:or 300 years ; the absorption of Egypt into the Roman Em
pire ; the closer commercial relations between peoples of the
East and of the West ; the gradual decline of paganism and
spread of Christianity, these events were of far-reaching
influence on the progress of the sciences, which then had their
home in Alexandria. Alexandria became a commercial and
intellectual emporium. Traders of all nations met in her
busy streets, and in her magnificent Library, museums, lecture-
halls, scholars from the East mingled with those of the
West; Greeks began to study older literatures and to com
pare them with their own. In consequence of this interchange
of ideas the Greek philosophy became fused with Oriental
THE GKEEKS. 57
The foundation of this science was laid by the illustrious
Hipparchus.
The Almagest is in 13 books. Chapter 9 of the first book
shows how to calculate tables of chords. The circle is divided
into 360 degrees, each of which is halved. The diameter is
divided into 120 divisions ; each of these into 60 parts, which
are again subdivided into 60 smaller parts. In Latin, these
parts were called partes minutes primce and paries mmutce
secundcB. Hence our names, minutes and seconds. 73 The
sexagesimal method of dividing the circle is of Babylonian
origin, and was known to Geminus and Hipparchus. But
Ptolemy s method of calculating chords seems "original with
him. He first proved the proposition, now appended to
Euclid VI. (D), that "the rectangle contained by the diag
onals of a quadrilateral figure inscribed in a circle is equal
to both the rectangles contained by its opposite sides." He
then shows how to find from the chords of two arcs the
chords of their sum and difference, and from the chord of any
arc that of its half. These theorems he applied to the calcu
lation of his tables of chords. The proofs of these theorems
are very pretty.
Another chapter of the first book in the Almagest is devoted
to trigonometry, and to spherical trigonometry in particular.
Ptolemy proved the lemma of Menelaus/ and also the c regula
sex quantitatum. Upon these propositions he built up his
trigonometry. The fundamental theorem of plane trigonome
try, that two sides of a triangle are to each other as the chords
of double the arcs measuring the angles opposite the two
sides, was not stated explicitly by him, but was contained
implicitly in other theorems. More complete are the proposi
tions in spherical trigonometry.
The fact that trigonometry was cultivated not for its own
sake, biit to aid astronomical inquiry, explains the rather
58 A HISTOKY OF MATHEMATICS.
startling fact that spherical trigonometry came to exist in a
developed state earlier than plane trigonometry.
The remaining books of the Almagest are on astronomy.
Ptolemy has written other works which have little or no bear
ing on mathematics, except one on geometry. Extracts from
this book, made by Proelus, indicate that Ptolemy did not
regard the parallel-axiom of Euclid as self-evident, and that
Ptolemy was the first of the long line of geometers from
ancient time down to our own who toiled in the vain attempt
to prove it.
Two prominent mathematicians of this time were Nicoma-
chus and Theon of Smyrna. Their favourite study was theory
of numbers. The investigations in this science culminated
later in the algebra of Diophantus. But no important geom
eter appeared after Ptolemy for 150 years. The only occupant
of this long gap was Sextus Julius Africanus, who wrote an
unimportant work on geometry applied to the art of war,
entitled Cestes.
Pappus, probably bora about 340 A.D., in Alexandria, was
the last great mathematician of the Alexandrian school. His
genius was inferior to that of Archimedes, Apollonius, and
Euclid, -who flourished over 500 years earlier. But living,
as he did, at a period when interest in geometry was declin
ing, he towered above his contemporaries "like the peak
of Teneriffa above the Atlantic." He is the author of a Com
mentary on the Almagest, a Commentary on JSucli& s JSlernents,
a Commentary on the Analemma of Diodorm, a writer of
whom nothing is known. All these works are lost. Proclus,
probably quoting from the Commentary on EiicUd, says that
Pappus objected to the statement that an, angle equal to a
right angle is always itself a right angle.
The only work of Pappus still extant is his Mathematical
Collections. This was originally in eight books, but the firsi
THE GBEBKS. 59
and portions of the second are now missing. The Mathemat
ical Collections seems to have been written by Pappus to supply
the geometers of his time with a succinct analysis of the most
difficult mathematical works and to facilitate the study of
them by explanatory lemmas. But these lemmas are selected
very freely, and frequently have little or no connection with the
subject on hand. However, he gives very accurate summaries
of the works of which he treats. The Mathematical Collections
is invaluable to us on account of the rich information it
gives on various treatises by the foremost Greek mathemati
cians, which are now lost. Mathematicians of the last century
considered it possible to restore lost works from the resume
by Pappus alone.
We shall now cite the more important of those theorems in
the Mathematical Collections which are supposed to be original
with Pappus. First of all ranks the elegant theorem re-dis
covered by Guldin, over 1000 years later, that pie volume
generated by the revolution of a plane curve which lies wholly
on one side of the axis, equals the area of the curve multiplied
by the circumference described by its centre of gravity.
Pappus proved also that the centre of gravity of a triangle is
that of another triangle whose vertices lie upon the sides of
the first and divide its three sides in the same ratio. In, the
fourth book are new and brilliant proposition^ on the quac|ra-
trix which indicate *&& intimate -acqnafitai!K^- Wifeii curvs^i
surfaces.^ He generates the quadratrix as follows : Let a
spiral line be drawn upon a right circular cylinder ; then the
perpendiculars to the axis of the cylinder drawn from each
point of* the spiral line form the surface of a screw. A plane
passed through one of these perpendiculars, making any con
venient angle with the base of the cylinder, cuts the screw-
surface in a curve, the orthogonal projection of which upon
the base is the quadratrix. A. second mode of generation is
60 A HISTORY OF MATHEMATICS.
no less admirable : If we make the spiral of Archimedes the
base of a right cylinder, and imagine a cone of revolution
having for its axis the side of the cylinder passing through
the initial point of the spiral, then this cone cuts the cylinder
in a curve of double curvature. The perpendiculars to the
axis drawn through every point in this curve form the surface
of a screw which Pappus here calls the plectoidal surface. A
plane passed through one of the perpendiculars at any con
venient angle cuts that surface in a curve whose orthogonal
projection upon the plane of the spiral is the required quadra-
trix. Pappus considers curves of double curvature still further.
He produces a spherical spiral by a point moving uniformly
along the circumference of a great circle of a sphere, while
the great circle itself revolves uniformly around its diameter.
He then finds the area of that portion of the surface of the
sphere determined by the spherical spiral, "a complanation
which claims the more lively admiration, if we consider that,
although the entire surface of the sphere was known since
Archimedes time, to measure portions thereof, such as spher
ical triangles, was then and for a long time afterwards an
unsolved problem." 8 A question which was brought into
prominence jby Descartes and Hewton is the "problem of
Pappus." ijGriven several straight lines in a plane, to find the
locus of a point such that when perpendiculars (or ? more
generally, straight lines at given angles) are drawn from it to
the given lines, the product of certain ones of them shall be in
a given ratio to the product of the remaining ones. It is
worth noticing that it was Pappus who first found the focus
of the parabola, suggested the iise of the directrix,! and pro
pounded the theory of the involution of points. He solved
the problem to draw through three points lying in the same
straight line, three straig% lines wiiich shaft form a triangle
inscribed in a given circle.* Prom the Mathematical Collections
THE GREEKS. 61
many more equally difficult theorems might be quoted which
are original with Pappus as far as we know. It ought to be
remarked; however, that he is known in three instances to
have copied theorems without giving due credit, and that he
may have done the same thing in other cases in which we
have no data by which to ascertain the real discoverer.
About the time of Pappus lived Theon of Alexandria. He
brought out an edition of Euclid s Elements with notes, which
he probably used as a text-book in his classes. His commen
tary on the Almagest is valuable for the many historical notices,
and especially for the specimens of Greek arithmetic which it
contains. Theon s daughter Hypatia, a woman celebrated for
her beauty and modesty, was the last Alexandrian teacher of
reputation, and is said to have been an abler philosopher and
mathematician than her father. Her notes on the works of
Diophantus and Apollonius have been lost. Her tragic death
in 415 A.D. is vividly described in Kingsley s Hypatia.
From now on, mathematics ceased to be cultivated in
Alexandria. The leading subject of men s thoughts was
Christian theology. Paganism disappeared, and with it pagan
learning. The Neo-Platonic school at Athens struggled on a
century longer. Proclus, Isidorus, and others kept up the
" golden chain of Platonic succession." Proclus, the successor
of Syrianus, at the Athenian school, wrote a commentary on
Euclid s Elements. We possess only that on the first book,
which is valuable for the information it contains on the
history of geometry. Damascius of Damascus, the pupil of
Tsidorus, is now believed to be the author of the fifteenth
book of Euclid. Another pupil of Isidorus was Eutocius of
Ascalon, the commentator of Apollonius and Archimedes.
Simplicius wrote a commentary on Aristotle s De Oodo. In
the year 529, Justinian, disapproving heathen learning, finally
closed by imperial edict the schools at Athens.
62 A HISTORY OF MATHEMATICS.
As a rule, the geometries of the last 500 years showed
a lack of creative power. They were commentators rather
than discoverers.
The principal characteristics of ancient geometry are :
(1) A wonderful clearness and defmiteness of its concepts
and an almost perfect logical rigour of its conclusions.
(2) A complete want of general principles and methods.
Ancient geometry is decidedly special Thus the Greeks
possessed no general method of drawing tangents. "The
determination of the tangents to the three conic sections did
not furnish any rational assistance for drawing the tangent to
any other new curve, such as the conchoid, the cissoid, etc." 35
In the demonstration of a theorem, there wore, for the ancient
geometers, as many different cases requiring separate proof
as there were different positions for the lines. The greatest
geometers considered it necessary to treat all possible cases
Independently of each other, and to prove each with equal
fulness. To devise methods by which the various eases could
all be disposed of by one stroke, was beyond the power of the
ancients. "If we compare a mathematical problem with a
huge rock, into the interior of which we desire to penetrate,
then the work of the Greek mathematicians appears to us like
that of a vigorous stonecutter who, with chisel and hammer,
begins with indefatigable perseverance, from without, to
crumble the rock slowly into fragments 5 the modern mathe
matician appears like an excellent minor, wlio first bores
through the rock some few passages, from which he then bursts
it into pieces with one powerful blast, and brings to light the
treasures within." I6
THE GREEKS. 63
GREEK ARITHMETIC.
G-reek mathematicians were in the habit of discriminating
between the science of numbers and the art of calculation.
The former they called arithmetical, the latter logistica. The
drawing of this distinction between the two was very natural
and proper. The difference between them is as marked as
that between theory and practice. Among the Sophists -the
art of calculation was a favourite study. Plato, on the other
hand, gave considerable attention to philosophical arithmetic,
but pronounced calculation a vulgar and childish art.
In sketching the history of Greek calculation, we shall first
give a brief account of the Greek mode of counting and of
writing numbers. Like the Egyptians and Eastern nations,
the earliest Greeks counted on their fingers or with pebbles.
In case of large numbers, the pebbles- were probably ar
ranged in parallel vertical lines. Pebbles on the first line
represented units, those on the second tens, those on the third
hundreds, and so on. Later, frames came into use/ in which
strings or wires took the place of lines. According to tra
dition, Pythagoras, who travelled in Egypt and, perhaps, in
India, first introduced this valuable instrument into Greece.
The abacus, ais it is called, existed among different peoples and
at different tim$s, in various stages of perfection. An abacus
is still employe! by the Chinese under the name of Sivan-pan.
We possess no specific information as to how the Greek abacus
looked or how it was used. Boethius says that the Pytha
goreans used with the abacus certain nine signs called apices,
which resembled in form the nine " Arabic numerals." But
the correctness of this assertion is subject to grave doubts.
The oldest Grecian numerical symbols were the so-called
Herodianic signs (after Herodianus, a Byzantine grammarian of
about 200 A.D., who describes them). These signs occur fre-
64 A HISTORY OF MATHEMATICS.
quently in Athenian inscriptions and are, on that account, now
generally called Attic. For some unknown reason these sym
bols were afterwards replaced by the alphabetic numerals, in
which the letters of the Greek alphabet were used, together
with three strange and antique letters & 9 , and 5), and the
symbol M. This change was decidedly for the worse, for the
old Attic numerals were less burdensome on the memory, inas
much as they contained fewer symbols and were better adapted
to show forth analogies in numerical operations. The follow
ing table shows the Greek alphabetic numerals and their
respective values :
1 2 8 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90
p<TTV<xV rcw ^/ a J /y etc.
100 200 300 400 500 600 700 800 900 1000 2000 3000
ft v
M M M etc.
10,000 20,000 30,000
It will be noticed that at 1000, the alphabet is begun over
again, but, to prevent confusion, a stroke is now placed before
the letter and generally somewhat bolow it, A horizontal line
drawn over a number served to distinguish it more readily
from words. The coefficient for M was sometimes placed
before or behind instead of over the M. Thus 43,678 was
written SM^yx 07 ?- ^ * s * * )e observed that the Greeks had no
zero.
Fractions were denoted by first writing the numerator
marked with an accent, then the denominator marked with
two accents and written twice. Thus, ly tO^nO" |^|. In case
of fractions having unity for the numerator, the a was omitted
and the denominator was written only once. Thus /x8" = -$%*
THE GBEBKS.
65
Greek writers seldom refer to calculation with alphabetic
numerals. Addition, subtraction, and even multiplication were
probably performed on the abacus. Expert mathematicians
may have used the symbols. Thus Eutocius, a commentator
of the sixth century after Christ, gives a great many multipli
cations of which the following is a specimen : 6
The operation is ex
plained sufficiently by the
modern numerals append
ed. In case of mixed
numbers, the process was
still more clumsy. Divis
265
265
8 a
MM
M cr/c e
40000, 12000, 1000
12000, 3600, 300
1000, 300, 25
70225
ions are found in Theon
of Alexandria s commen
tary on the Almagest. As
might be expected, the process is long and tedious.
We have seen in geometry that the more advanced mathe
maticians frequently had occasion to extract the square root.
Thus Archimedes in his Mensuration of the Circle gives a
large number of square roots. He states, for instance, that
V3 < l^y- and VS > f -f-f, but he gives no clue to the method
by which he obtained these approximations. It is not im
probable that the earlier Greek mathematicians found the
square root by trial only. Eutocius say^ that the method of
extracting it wsts given by Heron, Pappus, Theon, and other
commentators on the Almagest. Theon s is the only ancient
method known to us. It is the same as the one used nowa
days, except that sexagesimal fractions are employed in place
of our decimals. What the mode of procedure actually was
when sexagesimal fractions were not used, lias been the sub
ject of conjecture on the part of numerous modern writers. 17
Of interest, in connection with arithmetical symbolism, is
the Sand-Counter (Arenarius), an essay addressed by Archi-
66 A HISTORY OF MATHEMATICS.
medes to Gelon, king of Syracuse. In it Archimedes shows
that people are in error who think the sand cannot be counted,
or that if it can be counted, the number cannot be expressed
by arithmetical symbols. He shows that the number of grains
in a heap of sand not only as large as the whole earth, but as
large as the entire universe, can be arithmetically expressed.
Assuming that 10,000 grains of sand suffice to make a little
solid of the magnitude of a poppy-seed, and that the diameter
of a poppy-seed be not smaller than ^ part of a finger s
breadth; assuming further, that the diameter of the universe
(supposed to extend to the sun) be less than 10,000 diameters
of the earth, and that the latter be less than 1,000,000 stadia,
Archimedes finds a number which would exceed the number
of grains of sancl in the sphere of the universe. He goes on
even further. Supposing the universe to reach out to the fixed
stars, he finds that the sphere, having the distance from the
earth s centre to the fixed stars for its radius, would contain
a number of grains of sancl less than 1000 myriads of tho
eighth octad. In our notation, this number would be 10 (I3 or
1 with 63 ciphers after it. It can hardly be cioubtod that one
object which Archimedes had in view in making this calcula
tion was the improvement of the Greek symbolism. It is not
known whether he invented some short notation by which to
represent the above number or not.
We judge from fragments in the second book of "Pappus that
Apollonius proposed an improvement in the Greek method o
writing numbers, but its nature wo do not know. Thus we
see that the Greeks never possessed tho boon of a clear, com
prehensive symbolism. The honour of giving suoli to the world,
once for all, was reserved by tho irony of fate for a namdcBB
Indian of an unknown time, and we. know not whom to thank
for an invention of such importance to the general progress of
intelligence, 6
THE GREEKS. 75
suggestions of algebraic notation, and of the solution of
equations, then his Arithmetica is the earliest treatise on
algebra now extant. In this work is introduced the idea of
an algebraic equation expressed in algebraic symbols. His
treatment is purely analytical and completely divorced from
geometrical methods. He is, as far as we know, the first to
state that " a negative number multiplied by a negative num
ber gives a positive number." This is applied to the multi
plication of differences, such as (x l)(x 2). It must be
remarked, however, that Diophantus had no notion whatever
of negative numbers standing by themselves. All he knew
were differences, such as (2 x 10), in which 2 x could not be
smaller than 10 without leading to an absurdity. He appears
to be the first who could perform such operations as (x 1)
x(x 2) without reference to geometry. Such identities as
(a + 6) 2 = a 2 + 2 ab + 6 2 , which with Euclid appear in the ele
vated rank of geometric theorems, are with Diophantus the
simplest consequences of the algebraic laws of operation. His
sign for subtraction was ^/, for equality i. For unknown
quantities he had only one symbol, ?. He had no sign for
addition except juxtaposition. Diophantus used but few sym
bols, and sometimes ignored even these by describing an oper
ation in words when the symbol would have answered just
as well.
In the solution of simultaneous equations Diophantus adroitly
managed with only one symbol for the unknown quantities and
arrived at answers, most commonly, by the method of tentative
assumption, which consists in assigning to some of the unknown
quantities preliminary values, that satisfy only one or two of
the conditions. These values lead to expressions palpably
wrong, but which generally suggest some stratagem by which
r^lues can be secured satisfying all the conditions of the
>roblem.
76 A HISTORY OF MATHEMATICS.
Diophantus also solved determinate equations of the second
degree. We are ignorant of Ms method, for he nowhere goes
through with the whole process of solution, but merely states
the result. Thus, " 84 x 2 + 7 x = 7, whence x is found = ."
Notice he gives only one root. His failure to observe that a
quadratic equatioti has two roots, even when both roots are
positive, rather surprises us. It must be remembered, how
ever, that this same inability to perceive more than one out of
the several solutions to which a problem may point is common
to all Greek mathematicians. Another point to be observed
is that he never accepts as an answer a quantity which is
negative or irrational.
Diophantus devotes only the first book of his Arithmetica to
the solution of determinate equations. The remaining- books
extant treat mainly of indeterminate quadratic equations of the
form J.& 2 +JS& 4-0=?/ 2 , or of two simultaneous equations of the
same form. He considers several but not all the possible
cases which may arise in these equations. The opinion of
Nesselmann on the method of Diophantus, as stated by Gow,
is as follows : " (1) Indeterminate equations of the second
degree are treated completely only when the quadratic or
the absolute term is wanting: his solution of the equations
Ax*-\- (7= f and Ax 2 +Bx+ (7= ;?/ 2 is in many respects cramped.
(2) Eor the double equation of the second degree he has a
definite rule only when the quadratic term is wanting in both
expressions : even then his solution is not general. More com
plicated expressions occur only under specially favourable
circumstances." Thus, he solves B% + C = ?/ 2 , B$s + d 2 = y*.
The extraordinary ability of Diophantus lies rather in
another direction, namely, in his wonderful ingenuity to re
duce all sorts of equations to particular forms which ho knoW
Jiow to solve. Very great is the variety of problems considered!
The 130 problems found in the great work of Diophantus COB/-
THE ROMANS* 77
tain over 50 different classes of problems, which, are strung
together without any attempt at classification. But still more
multifarious than the problems are the solutions. General
methods are unknown to Diophantus. Each problem has its
own distinct method, which is often useless for the most
closely related problems. "It is, therefore, difficult for a
modern, after studying 100 Diophantine solutions, to solve
the 101st." 7
That which robs his work of much of its scientific value is
the fact that he always feels satisfied with one solution, though
his equation may admit of an indefinite number of values.
Another great defect is the absence of general methods. Mod
ern mathematicians, such as Euler, La Grange, Gauss, had to
begin the study of indeterminate analysis anew and received
no direct aid from Diophantus in the formulation of methods.
In spite of these defects we cannot fail to admire the work
for the wonderful ingenuity exhibited therein in the solution
of particular equations.
It is still an open question and one of great difficulty
whether Diophantus derived portions of his algebra from
Hindoo sources or not.
THE BOMANS.
Nowhere is the contrast .between the Greek and Eoman
mind shown forth more distinctly than in their attitude toward
the mathematical science. The sway of the Greek was a
flowering time for mathematics, but that of the Eoman a
period of sterility. In philosophy, poetry, and art the Eoman
was an imitator. But in mathematics he did not even rise to
the desire for imitation. The mathematical fruits of Greek
genius lay before him untasted. In him a science which had
78 A HISTOJEtY OF MATHEMATICS.
no direct bearing on practical life could awake no interest.
As a consequence, not only the higher geometry of Archimedes
and Apollonius, but even the Elements of Euclid, were en
tirely neglected. What little mathematics the Romans pos
sessed did not come from the Greeks, but from more ancient
sources. Exactly where and how it originated is a matter of
doubt. It seems most probable that the " Roman notation,"
as well as the practical geometry of the Romans, came from
the old Etruscans, who, at the earliest period to which our
knowledge of them extends, inhabited the district between the
Arno and Tiber.
Livy tells us that the Etruscans were in the habit of repre
senting the number of years elapsed, by driving yearly a nail
into the sanctuary of Minerva, and that the Romans continued
this practice. A less primitive mode of designating numbers,
presumably of Etruscan origin, was a notation resembling the
present " Roman notation." This system is noteworthy from
the fact that a principle is involved in it which is not met
with in any other ; namely, the principle of subtraction. If a
letter be placed before another of greater value, its value is
not to be added to, but subtracted from, that of the greater.
In the designation of large numbers a horizontal bar placed
over a letter was made to increase its value one thousand fold.
In fractions the Romans used the duodecimal system.
Of arithmetical calculations, the Romans cm ploy od three
different kinds : Reckoning on the fingers, upon the abacus,
and by tables prepared for the purpose, 8 Finger-symbolism
was known as early as the time of King Nuina, for he had
erected, says Pliny, a statue of the double-faced Janus, of
which the fingers indicated 305 (355?), the number of days in
a^ year. Many other passages from Roman authors point out
the use of the fingers as aids to calculation. In fact, a finger-
symbolism of practically the same form was in use not only in
THE KOMANS. 79
Bonie, but also in Greece and throughout the East, certainly
as early as the beginning of the Christian era, and continued
to be used in Europe during the Middle Ages. We possess no
knowledge as to where or when it was invented. The second
mode of calculation, by the abacus, was a subject of elemen
tary instruction in Borne. Passages in Eoman writers indicate
that the kind of abacus most commonlyuseiTwas" covered with
dust and then divided into columns by drawing straight lines.
Each column was supplied with pebbles (calculi, whence cal-
culare 3 and calculate 3 ) which served for calculation. Addi
tions and subtractions could be performed on the abacus quite
easily, but in multiplication the abacus could be used only for
adding the particular products^ and in division for performing
the subtractions occurring in the process. Doubtless at this
point recourse was made to mental operations and to the mul
tiplication table. Possibly finger-multiplication may also have
been used. But the multiplication of large numbers must, by
either method, have been beyond the power of the ordinary
arithmetician. To obviate this difficulty, the arithmetical
tables mentioned above were used, from which the desired
products could be copied at once. Tables of this kind were
prepared by Victorius of Aquitania. His tables contain a
peculiar notation for fractions, which continued in use through
out the Middle Ages. Victorius is best known for his canon
pascJialis, a rtiterftrr finding the correct date for Easter, which
he published in 457 A.D.
Payments of interest and problems in interest were very old
among the Bomans. The Roman laws of inheritance gave
rise to numerous arithmetical examples. Especially unique is
the following : A dying man wills that, if his wife, being with
child, gives birth to a son, the son shall receive f and she -j- of
his estates ; but if a daughter is born, she shall receive $ and
his wife -|. It happens that twins are born, a boy and a girl.
80 A HISTORY OF MATHEMATICS.
How shall the estates be divided so as to satisfy the will?
The celebrated Eoman jurist, Salvianus Julianus, decided that
the estates shall be divided into seven equal p&its, of which
the. son receives four, the wife two, the daughter one.
We next consider Eoman geometry. He who expects to
find in Koine a science of geometry, with definitions, axioms,,
theorems, and proofs arranged in logical order, will be disap
pointed. The only geometry known was a practical geometry,
which, like the old Egyptian, consisted only of empirical rules.
This practical geometry was employed in surveying. Treatises
thereon have come down to us, compiled by the Roman sur
veyors, called agrimensores or gromatici. One would naturally
expect rules to be clearly formulated. But no ; they are left
to be abstracted by the reader from a mass of numerical exam
ples. "The total impression is as though the Eoman gromatic
were thousands of years older than Greek geometry, and as
though a deluge were lying between the two." Some of their
rules were probably inherited from the " Etruscans, but others
are identical with those of Heron. { Among the latter is that
for finding the area of a triangle from its sides and the approx
imate formula, -|-| a 2 , for the area of equilateral triangles (a
being one of the sides) . But the latter area was also calculated
by the formulas -J-(a 2 +a) and -|a 2 , the first of which was
unknown to Heron. Probably the expression. |a 2 was derived
from the Egyptian formula ii-r. i- for the determina-
2 2t
tion of the surface of a quadrilateral. This Egyptian formula
was used by the Romans for finding the area, not only of rec
tangles, but of any quadrilaterals whatever. Indeed, the groma-
tici considered it eveii sufficiently accurate to determine the
areas of cities, laid out irregularly, simply by measuring their
circumferences. 7 Whatever Egyptian geometry the Romans
possessed was transplanted across th Mediterranean at the
THE KOMANS. 81
time of Julius Ccesar, who ordered a survey of the whole
empire iso secure an equitable mode of taxation. Ceesar also
.reformed Hh e calendar, and, for that purpose, drew from
Egyptian learning. He secured the services of the Alexan
drian astronomer, Sosigenes.
In the fifth century, the Western E/oman Empire was fast
falling to pieces. Three great branches Spain, Gaul, and
the province of Africa broke off from the decaying trunk.
In 476 ; the Western Empire passed away, and the Visigothic
chief, Odoacer, became king. Soon after, Italy was conquered
by the Ostrogoths under . Theodoric. It is remarkable that
this very period of political humiliation should be the one
during which Greek science was studied in Italy most zeal
ously. School-books began to be compiled from the elements
of Greek authors. These compilations are very deficient, but
are of absorbing interest, from the fact that, down to .the
twelfth century, they were the only sources of mathematical
knowledge in the Occident. Eoremost among these writers is
BoetMus (died 524). At first he was a great favourite of King
Theodoric, but later, being charged by envious courtiers with
treason, he was imprisoned, and at last decapitated. While
in prison he wrote On the Consolations of Philosophy. As a
mathematician, Boethius was a Brobdingnagian among Eoman
scholars, but a Liliputian by the side of Greek masters. He
wrote an In stitutis Arithmetica, which is essentially a transla
tion of the arithmetic of ISTicomachus, and a Geometry in
several books. Some of the most beautiful results of Mco-
machus are omitted in Boethius arithmetic. The first book
on geometry is an extract from Euclid s Elements, which con
tains, in addition to definitions, postulates, and axioms, the
theorems in the first three books, without proofs. How can
this omission of proofs be accounted for ? It has been argued
by some that Boethius possessed an incomplete Greek copy of
82 A HISTOBY OF MATHEMATICS.
the Elements; by others, that he had Theon s edition before
him, and believed that only the theorems came from Euclid,
while the proofs were supplied by Theon. The second book,
as also other books on geometry attributed to Boethius,
teaches, from numerical examples, the mensuration of plane
figures after the fashion of the agriniensores.
A celebrated portion in the geometry of Boethius is that
pertaining to an abacus, which he attributes to the Pythago
reans. A considerable improvement on the old abacus is
there introduced. Pebbles are discarded, and apices (probably
small cones) are used. Upon each of these apices is drawn
a numeral giving it some value below 10. The names of
these numerals are pure Arabic, or nearly so, but are added,
apparently, by a later hand. These figures are obviously the
parents of our modern "Arabic" numerals. The is not
mentioned by Boethius in the text. These numerals bear
striking resemblance to the Gubar-numerals of the West-
Arabs, which are admittedly of Indian origin. These facts
have given rise to an endless controversy. Some contended
that Pythagoras was in India, and from there brought the
nine numerals to Greece, where the Pythagoreans used them
secretly. This hypothesis has been generally abandoned, for
it is not certain that Pythagoras or any disciple of his ever
was in India, nor is there any evidence in any Greek author,
that the apices were known to the Greeks, or that numeral
signs of any sort were used by them with the abacus. It is
improbable, moreover, that the Indian signs, from which the
apices are derived, are so old as the time of Pythagoras.
A second theory is that the Geometry attributed to Boethius
is a forgery ; that it is not older than the tenth, or possibly
the ninth, century, and that the apices are derived from the
Arabs. This theory is based on contradictions between pas
sages in the AritJimetica and others in the Geometry. But
THE ROMANS. 83
there is an Encyclopaedia written by Gassiodorius (died about
570) in which both the arithmetic and geometry of Boethius
are mentioned. There appears to be no good reason for doubt
ing the trustworthiness of this passage in the Encyclopaedia.
{L third theory (Woepcke s) is that the Alexandrians either
directly or indirectly obtained the nine numerals from the
Hindoos, about the second century A.D., and gave them to
the Romans on the one hand, and to the Western Arabs
on the other. / This explanation is the most plausible.
MIDDLE AGES.
THE HINDOOS.
THE first people who distinguished themselves in mathe
matical research, after the time of the ancient Greeks, belonged,
like them, to the Aryan race. It was, however, not a Euro
pean, but an Asiatic nation, and had its seat in far-off India.
Unlike the Greek, Indian society was fixed into castes. The
only castes enjoying the privilege and leisure for advanced
study and thinking were the Brahmins, whose prime business
was religion and philosophy, and the IZshatriyas, who attended
to war and government.
Of the development of Hindoo mathematics we know but
little. A few manuscripts bear testimony that the Indians
had climbed to a lofty height, but their path of ascent is no
longer traceable. It would seem that Greek mathematics grew
up under more favourable conditions than the Hindoo, for in
Greece it attained an independent existence, and was studied
for its own sake, while Hindoo mathematics always remained
merely a servant to astronomy. Furthermore, in Greece
mathematics was a science of the people, free to be cultivated
by all who had a liking for it ; in India, as in Egypt, it was in
the hands chiefly of the priests. Again, the Indians were in
the habit of putting into verse all mathematical results they
obtained, and of clothing them in obscure and mystic language,
84
THE HINDOOS. 85
which, though "well adapted to aid the memory of him who
already understood the subject, was often unintelligible to the
uninitiated. Although the great Hindoo mathematicians
doubtless reasoned out most or all of their discoveries, yet
they were not in the habit of preserving the proofs, so that
the naked theorems and processes of operation are all that
have come down to our time. Very different in these respects
were the Greeks. Obscurity of language was generally
avoided, and proofs belonged to ihe stock of knowledge quite
as much as/fthe theorems themselves. Very striking was the
difference in the bent of mind of the Hindoo and Greek ; for,
while the Greek mind was pre-eminently geometrical, the
Indian was first of all arithmetical The Hindoo dealt with
number, the Greek with form. Numerical symbolism, the
science of numbers, and algebra attained in India far greater
perfection than they had previously reached in Greece. On
the other hand, we believe that thei^e was little or no geom
etry in India of which the source may not be traced back to
Greece. Hindoo trigonometry might possibly be mentioned
as an exception, BuT it rested on arithmetic more than on
geometry.
An interesting but difficult task is the tracing of the rela
tion between Hindoo and Greek mathematics. It is well
known that more or less trade was carried on be l veen Greece
and India from early times. After Egypt had become a
Eoman province, a more lively commercial intercourse sprang
up between Rome and India, by way of Alexandria. A priori,
it does not seem improbable, that with the traffic of merchan
dise there should also be an interchange of ideas. That
communications of thought from the Hindoos to the Alexan
drians actually did take place, is evident from the fact that
. certain philosophic and theologic teachings of the Manicheans,
Teo-Platomsts, Gnostics, show unmistakable likeness to
86 A HISTORY OF MATHEMATICS.
Indian tenets. Scientific facts passed also from Alexandria
to India. This is shown plainly by the Greek origin of some
of the technical terms used by the Hindoos. Hindoo astron
omy was influenced by Greek astronomy. Most of the geo
metrical knowledge which they possessed is traceable to
Alexandria, and to the writings of Heron in particular. In
algebra there was, probably, a mutual giving and receiving.
We suspect that Diophantus got the first glimpses of algebraic
knowledge from India. On the other hand, evidences have
been found of Greek algebra among the Brahmins. The
earliest knowledge of algebra in India may possibly have been
of Babylonian origin. When we consider that Hindoo scien
tists looked upon arithmetic and algebra merely as tools
useful in astronomical research, there appears deep irony in
the fact that these secondary branches were after all the only
ones in which they won real distinction, while in their pet
science of astronomy they displayed an inaptitude to observe,
to collect facts, and to make inductive investigations.
We shall now proceed to enumerate the names of the
leading Hindoo mathematicians, and then to review briefly
Indian mathematics. We shall consider the science only in
its complete state, for our data are not sufficient to trace the
history of the development of methods. Of the great Indian
mathematicians, or rather, astronomers, for India had no
mathematicians proper, Aryabhatta is the earliest. He was
born 476 A.r>., at Pataliputra, on the upper Ganges. His
celebrity rests on a work entitled Aryabhattiyam, of which
the third chapter is devoted to mathematics. About one
hundred years later, mathematics in India reached the highest
mark. At that time flourished Brahraagupta (born 598). In
628 he wrote his Brahma-sphutOrSiddhanta ("The Revised Sys
tem of Brahma"), of which the twelfth and eighteenth chapters
belong to mathematics. To the fourth or fifth century belongs
THE HINDOOS. 87
an anonymous astronomical work, called Surya-siddhanta
("Knowledge from the Sun"), which by native authorities
was ranked second only to -the Brahma-siddJianta, but is of in
terest to us merely as furnishing evidence that Greek science
influenced Indian science even before the time of Aryabhatta.
The following centuries produced only two names of impor
tance; namely, Cridhara, who wrote a Ganita-sam ("Quintes
sence of Calculation 3 ), and Padmanabha, the author of an
algebra. The science seems to have made but little progress
at this time ; for a work entitled Siddhantaciromani ("Diadem
of an Astronomical System "), written by Bhaskara Acarya in
1150, stands little higher than that of Brahmagupta, written
over 500 years earlier. The two most important mathematical
chapters in this work are the Lilavati ( = "the beautiful," i.e.
the noble science) and Viga-ganita (= "root-extraction"), de
voted to arithmetic and algebra. From now on, the Hindoos
in the Brahmin schools seemed to content themselves with
studying the masterpieces of their predecessors. Scientific
intelligence decreases continually, and in modern times a very
deficient Arabic work of the sixteenth century has been held
in great authority/
The mathematical chapters of the BraJima-siddhanta and
Siddhantaciromani were translated into English by H. T.
Colebrooke, London, 1817. The Surya-siddhanta was trans
lated by E. Burgess, and annotated by W. D. Whitney, New
Haven, Conn., 1860.
r The grandest achievement of the Hindoos and the one
which, of all mathematical inventions, has contributed most
to the general progress of intelligence, is the invention of
the principle of position in writing numbers. Generally we
speak of our notation as the " Arabic " notation, but it should
be called the "Hindoo" notation, for the Arabs borrowed it
\rom the Hindoos. That the invention of this notation was
88 A HISTORY OF MATHEMATICS.
not so easy as we might suppose at first thought, may be
inferred from the fact that, of other nations, not even the
keen-minded Greeks possessed one- like it. We inquire, -who
invented this ideal symbolism, and when? But we know
neither the inventor nor the time of invention. That our
system of notation is of Indian origin is the only point of
which we are certain. From the evolution of ideas in general
we may safely infer that our notation did not spring into
existence a completely armed Minerva from the head of
Jupiter. The nine figures for writing the units are supposed
to have been introduced earliest, and the sign of zero and the
principle of position to be of later origin. This view receives
support from the fact that on the island of Ceylon a notation
resembling the Hindoo, but without the zero has been pre
served. We know that Buddhism and Indian culture were
transplanted to Ceylon about the third century after Christ,
and that this culture remained stationary there, while it made
progress on the continent. It seems highly probable, then,
that the numerals of Ceylon are the old, imperfect numerals
of India. In Ceylon, nine figures were used for the units,
nine others for the tens, one for 100, and also one for 1000.
These 20 characters enabled them to write all the numbers up
to 9999. Thus, 8725 would have been written with six signs,
representing the following numbers : 8, 1000, 7, 100, 20, 5.
These Singhalesian signs, like the old Hindoo numerals, are
supposed originally to have been the initial letters of the corre
sponding numeral adjectives. There is a marked resemblance
between the notation of Ceylon and the one used by Aryabhatta
in the first chapter of his work, and there only. Although the
zero and the principle of position were unknown to the scholars
of Ceylon, they were probably known to Aryabhatta; for, in
the second chapter, he gives directions for extracting the square
and cube roots, which seem to indicate a knowledge of them.
THE HINDOOS. 89
It would appear that the zero and the accompanying principle
of position were introduced about the time of Aryabhatta.
The se are the inventions which give the Hindoo system its
great superiority, its admirable perfection.
There appear to have been several notations in use in
different parts of India, which differed, not in principle, but
merely in the forms of the signs employed. Of interest is
also a symbolical system of position^ in which the figures
generally were not expressed by numerical adjectives, but by
objects suggesting the particular numbers in question. Thus,
for 1 were used the words moon, Brahma, Creator, or form;
for 4, the words Feda, (because it is divided into four parts)
or ocean, etc. The following example, taken from the Surya-
siddJianta, illustrates the idea. The number 1,577,917,828 is
expressed from right to left as follows: Vasu (a class of 8
gods) + two + eight -f mountains (the 7 mountain-chains)
+ form + digits (the 9 digits) + seven + mountains + lunar
days (half of which equal 15). The use of such notations
made it possible to represent a number in several different
ways. This greatly facilitated the framing of verses con
taining arithmetical rules or scientific constants, which could
thus be more easily remembered.
At an early period the Hindoos exhibited great skill in
calculating, even with large numbers. Thus, they tell us of
an examination to which Buddha, the reformer of the Indian
religion, had to submit, when a youth, in order to win the
maiden he loved. In arithmetic, after having astonished his
examiners by naming all the periods of numbers up to the
53d, he was asked whether he could determine the number
of primary atoms which, when placed one against the other,
would form a line one mile in length. Buddha found the
required answer in this way : 7 primary atoms make a very
minute grain of dust, 7 of these make a minute grain of dust,
90 A HISTOBY OF MATHEMATICS.
7 of tJiese a grain of dust whirled up by the wind, and so on.
Thus he proceeded, step by step, until he finally reached the
length of a mile. The multiplication of all the factors gave
for the multitude of primary atoms in a mile a number con
sisting of 15 digits. This problem reminds one of the Sand-
Counter 7 of Archimedes.
After the numerical symbolism had been perfected, figuring
was made much easier. Many of the Indian modes of
operation differ from ours. The Hindoos were generally
inclined to follow the motion from left to right, as in writing.
Thus, they added the left-hand columns first, and made the
necessary corrections as they proceeded. 3?or instance, they
would have added 254 and 663 thus : 2 + 6 = 8, 5 + 6 = 11,
which changes 8 into 9, 4 4- 3 = 7. Hence the sum 917. In
subtraction they had two methods. Thus in 821 348 they
would say, 8 from 11 = 3, 4 from 11 = 7, 3 from 7 = 4. Or
they would say, 8 from 11*= 3, 5 from 12 = 7, 4 from 8=4.
In multiplication of a number by another of only one digit, say
569 by 5, they generally said, 5-5 = 25, 5-6 = 30, which
changes 25 into 28, 5-9 = 45, hence the must be increased by
4. The product is 2845. In the multiplication with each
other of many-figured numbers, they first multiplied, in the
manner just indicated, with the left-hand digit of the multi
plier, which was written above the multiplicand, and placed
the product above the multiplier. On multiplying with the
next digit of the multiplier, the product was not placed in
a new row, as with us, but the first product obtained was
corrected, as the process continued, by erasing, whenever
necessary, the old digits, and replacing them by new ones,
until finally the whole product was obtained. Wo who possess
the modern luxuries of pencil and paper, would not be likely
to fall in love with this Hindoo method. But the Indians
wrote " with a cane-pen upon a small blackboard with a white,
THE HINDOOS. 93
Passing now to algebra, we shall first take up the symbols
of operation. Addition was indicated simply by juxtaposition
as in Diophantine algebra ; subtraction, by placing a dot over
the subtrahend ; multiplication, by putting after the factors
bha, the abbreviation of the word bhavita, "the product";
division, by placing the divisor beneath the dividend ; square-
root, by writing Tea, from the word Tcarana (irrational), before
the quantity. The unknown quantity was called by Brahma-
gupta ydvattdvat (quantum tantum) . When several unknown
quantities occurred, he gave, unlike Diophantus, to each a
distinct name and symbol. The first unknown was designated
by the general term "unknown quantity." The rest were
distinguished by names of colours, as the black, blue, yellow,
red, or green unknown. The initial syllable of each word
constituted the symbol for the respective unknown quantity.
Thus yd, me^nt x; Ted (from "kdla ka^ black) meant yj yd Jed
bha, " x times y " ; Tea 15 Tea 10 3 " Vl5 VlO."
The Indians were the first to recognise the existence of
absolutely negative quantities. They brought out the differ
ence between positive and negative quantities by attaching to
the one the idea of possession/ to the other that of debts/
The conception also of opposite directions on a line, as an
interpretation of + and quantities, was not foreign to them.
They advanced beyond Diophantus in observing that a quad
ratic has always two roots. Thus Bhaskara gives x = 50 and
x= 5 for the roots of x 2 - 45 x = 250. "But," says he,
"the second value is in this case not to be taken, for it is
inadequate ; people do not approve of negative roots." Com
mentators speak of this as if negative roots were seen, but not
admitted.
Another important generalisation, says Hankel, was this,
that the Hindoos never confined their arithmetical operations
to rational numbers. For instance, Bhaskara showed how,
94 A HISTOBY OF MATHEMATICS.
, ,, i . T . - c& 2 2> . . /a - V a
by the formula V a + V5 ==^--1-- -- (--y
the square root of the sum of rational and irrational numbers
could be found. The Hindoos never discerned the dividing
line between numbers and magnitudes, set up by the Greeks,
which, though the product of a scientific spirit, greatly re
tarded the progress of mathematics. They passed from mag
nitudes to numbers and from numbers to magnitudes without
anticipating that gap which to a sharply discriminating mind
exists between the continuous and discontinuous. Yet by
doing so the Indians greatly aided the general progress of.
mathematics. " Indeed, if one understands by algebra the
application of arithmetical operations to complex magnitudes
of all sorts, whether rational or irrational numbers or space-
magnitudes, then the learned Brahmins of Hindostan are the
real inventors of algebra." 7
Let us now examine more closely the Indian algebra. In
extracting the square and cube roots they used the formulas
(a + Z>) 2 = a 2 + 2 ab + 5 2 and (a + &)*= ^ + 3 a s 6 + 3 ab 2 + W.
In this connection Aryabhatta speaks of dividing a number
into periods of two and three digits. From this we infer that
the principle of position and the zero in the numeral notation
were already known to him. In figuring with zeros, a state
ment of Bhaskara is interesting. A fraction whose denomi
nator is zero, says he, admits^of,^,,, alteration, though much be
added or subtracted. Indeed, in the same way, no change
taEes" place "motile Infinite and immutable Deity when worlds
are destroyed or created, even though numerous orders of beings
be taken, up or brought forth. Though in this he apparently
evinces clear mathematical notions, yet in other places he
jifakes a complete failure in figuring with fractions of zero
denominator.
In the Hindoo solutions of determinate equations, Cantor
THE HINDOOS. 95
thinks he can see traces of Diophantine methods. Some
technical terms betray their Greek origin. Even if it be true
that the Indians borrowed from the Greeks, they deserve great
credit for improving and generalising the solutions of linear
and quadratic equations. Bhaskara advances far beyond the
Greeks and even beyond Brahmagupta when he says that
"the square of a positive, as also of a negative number,
is positive; that the square root of a positive number is
twofold, positive and negative. There is no square root
of a negative number, for it is not a square." Of equa
tions of higher degrees, the Indians succeeded in solving
only some special cases in which both sides of the equation
could be made perfect powers by the addition of certain
terms to each.
Incomparably greater progress than in the solution of deter
minate equations was made by the Hindoos in the treatment
of indeterminate equations. Indeterminate analysis was a
subject to which the Hindoo mind showed a happy adaptation.
We have seen that this very subject was a favourite with Dio-
phantus, and that his ingenuity was almost inexhaustible in
devising solutions for particular cases. But the glory of
having invented general methods in this most subtle branch
of mathematics belongs to the Indians. The Hindoo indeter
minate analysis differs from the Greek not only in method,
but also in aim. The object of the former was to find all
possible integral solutions. Greek analysis, on the other hand,
demanded not necessarily integral, but simply rational answers.
Diophantus was content with a single solution ; the Hindoos
endeavoured to find all solutions possible. Aryabhatta gives
solutions in integers to linear equations of the form ax by=c,
where a, 6, c are* integers. The rule employed is called the
pulveriser. or this, as for most other rules, the Indians give
no proof. Their solution is essentially the same as the one of
96 A HISTORY OF MATHEMATICS.
Euler. Euler s process of reducing ~ ,to a continued fraction-
amounts to the same as the Hindoo process of finding the
greatest common divisor of a and b by division. This is fre
quently called the Diophantine method. Hankel protests
against this name, on the ground that Diophantus not only
never knew the method, but did not even aim at solutions
purely integral. 7 These equations probably grew out of prob
lems in astronomy. They were applied, for instance, to
determine the time when a certain constellation of the planets
would occur in the heavens.
Passing by the subject of linear equations with more than
two unknown quantities, we come to indeterminate quadratic
equations. In the solution of xy = ax + "by + c, they applied
the method re-invented later by Euler, of decomposing (ab + c)
into the product of two integers m - n and of placing a; = m + b
and y = n + a.
Remarkable is the Hindoo solution of the quadratic equa
tion cy 2 = ace 2 + b. With great keenness of intellect they
recognised in the special case 2/ 2 = a& 2 + l a fundamental
problem in indeterminate quadratics. They solved it by the
cyclic method. " It consists," says De Morgan, " in a rule for
finding an indefinite number of solutions of y 2 = ay? + 1 (a be
ing an integer which is not a square), by means of one solution
given or found, and of feeling for one solution by making a
solution of t/ 2 = ay? + b give a solution of y* = ace 2 + W. It
amounts to the following theorem : If p and q be one set of
values of x and y in y 2 = ax 2 + b and p 1 and q 1 the same or
another set, then qp + pq and app* + qq are values of a? and y
in 2/ 2 = ace 2 + 6 2 . JYom this it is obvious that one solution of
2/ 2 = ay? + 1 may be made to give any number, and that if,
taking b at pleasure, t/ 2 == aa? + b 2 can be solved so that x and y
are divisible by b, then one preliminary solution of y* = ax* + 1
THE ARABS. 101
in Spain. Astounding as was the grand march of conquest by
the Arabs, still more so was the ease with which they put
aside their former nomadic life, adopted a higher civilisation,
and assumed the sovereignty over cultivated peoples. Arabic
was made the written language throughout the conquered
lands. With the rule of the Abbasides in the East began a
new period in the history of learning. The capital, Bagdad,
situated on the Euphrates, lay half-way between two old
centres of scientific thought, India in the East, and Greece
in the West. The Arabs were destined to be the custodians
of the torch of Greek and Indian science, to keep it ablaze
during the period of confusion and chaos in the Occident, and
afterwards to pass it over to the Europeans. Thus science
passed from Aryan to Semitic races, and then back again
to the Aryan. The Mohammedans have added but little to
the knowledge in mathematics which they received. They
now and then explored a small region to which the path had
been previously pointed out, but they were quite incapable of
discovering new fields. Even the more elevated regions in
which the Hellenes and Hindoos delighted to wander
namely, the Greek conic sections and the Indian indeterminate
analysis were seldom entered upon by the Arabs. They
were less of a speculative, and more of a practical turn of
mind.
The Abbasides at Bagdad encouraged the introduction of
the sciences by inviting able specialists to their court, irre
spective of nationality or religious belief. Medicine and
astronomy were their favourite sciences. Thus Haroun-al-
Baschid, the most distinguished Saracen ruler, drew Indian
physicians to Bagdad. In the year 772 there came to the
3ourt of Caliph Almansur a Hindoo astronomer with astronom
ical tables which were ordered to be translated into Arabic.
These tables, known by the Arabs as the SindMnd, and
102 A HISTORY OF MATHEMATICS.
probably taken from the Brahma-sphuta-siddhanta of Brahma-
gupta, stood in great authority. They contained the important
Hindoo table of sines.
Doubtless at this time, and along with these astronomical
tables, the Hindoo numerals, with the zero and the principle
of position, were introduced among the Saracens. Before the
time of Mohammed the Arabs had no numerals. Numbers
were written out in words. Later, the numerous computations
connected with the financial administration over the conquered
lands made a short symbolism indispensable. In some locali
ties, the numerals of the more civilised conquered nations
were used for a time. Thus in Syria, the Greek notation was
retained; in Egypt, the Coptic. In some cases, the numeral
adjectives may have been abbreviated in writing. The Diwani-
numeralSj found in an Arabic-Persian dictionary, are supposed
to be such abbreviations. Gradually it became the practice to
employ the 28 Arabic letters of the alphabet for numerals, in
analogy to the Greek system. This notation was in turn
superseded by the Hindoo notation, which quite early was
adopted by merchants, and also by writers on arithmetic. Its
superiority was so universally recognised, that it had no rival,
except in astronomy, where the alphabetic notation continued
to be used. Here the alphabetic notation offered no great
disadvantage, since in the sexagesimal arithmetic, taken from
the Almagest, numbers of generally only one or two places
had to be written. 7
As regards the form of the so-called Arabic numerals, tlie
statement of the Arabic writer Albiruni (died 1039), who
spent many years in India, is of interest. He says that tlie
shape of tlie numerals, as also of the letters in India, differed
in different localities, and that the Arabs selected from the
various forms the most suitable. An Arabian astronomer
says there was among people much difference in the use of
THE AEABS. 103
symbols, especially of those for 5, 6, 7, and 8. The symbols
used by the Arabs can be traced back to the tenth century.
We find material differences between those used by the
Saracens in the East and those used in the West. But
most surprising is the fact that the symbols of both the East
and of the West Arabs deviate so extraordinarily from the
Hindoo Devanagari numerals (= divine numerals) of to-day,
and that they resemble much more closely the apices of
the Eoman writer Boethius. This strange similarity on the
one hand, and dissimilarity on the other, is difficult to explain.
The most plausible theory is the one of Woepcke: (1) that
about the second century after Christ, before the zero had
been invented, the Indian numerals were brought to Alexan
dria, whence they spread to Eome and also to West Africa ;
(2) that in the eighth century, after the notation in India had
been already much modified and perfected by the invention of
the zero, the Arabs at Bagdad got it from the Hindoos ; (3) that
the Arabs of the West borrowed the Columbus-egg, the zero,
from those in the East, but retained the old forms of the nine
numerals, if for no other reason, simply to be contrary to their
political enemies of the East; (4) that the old forms were
remembered by the West-Arabs to be of Indian origin, and
were hence called Ghtbar-nuwierdls ( = dust-numerals, in mem
ory of the Brahmin practice of reckoning on tablets strewn
with dust or sand; 1 (5) that, since the eighth century, the
numerals in India underwent further changes, and assumed
the greatly modified forms of the modern Devanagari-numer-
als. 3 This is rather a bold theory, but, whether true or not,
it explains better than any other yet propounded, the relations
between the apices, the Gubar, the East-Arabic, and Devana
gari numerals.
It has been mentioned that in 772 the Indian SiddJianta was
brought to Bagdad and there translated into Arabic. There
104 A HISTORY OF MATHEMATICS.
is no evidence that any intercourse existed between Arabic
and Indian astronomers either before or after this time, ex
cepting the travels of Albiruni. But we should be very slow
to deny the probability that more extended communications
actually did take place.
Better informed are we regarding the way in which Greek
science, in successive waves, dashed upon and penetrated Arabic
soil. In Syria the sciences, especially philosophy and medi
cine, were cultivated by Greek Christians. Celebrated were the
schools at Antioch and Emesa, and, first of all, the flourishing
E"estorian school at Edessa. Erom Syria, Greek physicians
and scholars were called to Bagdad. Translations of works
from the Greek began to be made. A large number of Greek
manuscripts were secured by Caliph Al Mamun (813-883) from
the emperor in Constantinople and were turned over to Syria.
The successors of Al Mamun continued the work so auspic
iously begun, until, at the beginning of the tenth century, the
more important philosophic, medical, mathematical, and as
tronomical works of the Greeks could all be read in the Arabic
tongue. The translations of mathematical works must have
been very deficient at first, as it was evidently difficult to
secure translators who were masters of both the Greek and
Arabic and at the same time proficient in mathematics. The
translations had to be revised again and again before they
were satisfactory. The first Greek authors made to speak in
Arabic were Euclid and Ptolemasus. * This was accomplished
during the reign of the famous Haroun-al-Easchid. A revised
translation of Euclid s Elements was ordered by Al Mamun.
As this reyision still contained numerous errors, a new trans
lation was made, either by the learned Honein ben Ishak, or
by his son, Ishak ben Honein. To the thirteen books of the
Elements were added the fourteenth, written by Hypsicles,
and the fifteenth by Damascius. But it remained for Tabit
THE ABABS. 105
ben Korra to bring forth an Arabic Euclid satisfying every
need. Still greater difficulty was experienced in securing an
intelligible translation of the Almagest. Among other impor
tant translations into Arabic were the works of Apollonius,
Archimedes, Heron, and Diophantus. Thus we see that in
the course of one century the Arabs gained access to the vast
treasures of Greek science. Having been little accustomed to
abstract thought, we need not marvel if, during the ninth cen
tury, all their energy was exhausted merely in appropriating
the foreign material. No attempts were made at original
work in mathematics until the next century.
In astronomy, on the other hand, great activity in original
research existed as early as the ninth century. The religious
observances demanded by Mohammedanism presented to as
tronomers several practical problems. The Moslem dominions
being of such enormous extent, it remained in some localities
for the astronomer to determine which way the "Believer"
must turn during prayer that he may be facing Mecca. The
prayers and ablutions had to take place at definite hours dur
ing the day and night. This led to more accurate determina
tions of time. To fix the exact date for the Mohammedan
feasts it became necessary to observe more closely the motions
of the moon. In addition to all this, the old Oriental supersti
tion that extraordinary occurrences in the heavens in some
mysterious way affect the progress of human affairs added
increased interest to the prediction of eclipses. 7
For these reasons considerable progress was made. Astro
nomical tables and instruments were perfected, observatories
erected, and a connected series of observations instituted. This
intense love for astronomy and astrology continued during the
whole Arabic scientific period. As in India, so here, we hardly
ever find a man exclusively devoted to pure mathematics. Most
of the so-called mathematicians were first of all astronomers.
106 A HISTORY Otf MATHEMATICS.
The first notable author of mathematical books was Moham
med ben Musa Hovarezmi, who lived during the reign of Caliph
Al Mamun (814-833) . He was engaged by the caliph in mak
ing extracts from the SindMnd, in revising the tablets of Ptole-
maeus, in taking observations at Bagdad and Damascus, and in
measuring a degree of the earth s meridian. Important to us
is his work on algebra and arithmetic. The portion on arith
metic-is not extant in the original, and it was not till 1857
that a Latin translation of it was found. It begins thus:
" Spoken has Algoritmi. Let us give deserved praise to God,
our leader and defender." Here the name of the author, Ho-
varezmi, has passed into Algoritmi, from which comes our
modern word algorithm, signifying the art of computing in
any particular way. The arithmetic of Hovarezmi, being
based on the principle of position and the Hindoo method of
calculation, "excels/ 7 says an Arabic writer, "all others in
brevity and easiness, and exhibits the Hindoo intellect and
sagacity in the grandest inventions." This book was followed
by a large number of arithmetics by later authors, which dif
fered from the earlier ones chiefly in the greater variety of
methods. Arabian arithmetics generally contained the four
operations with integers and fractions, modelled after the
Indian processes. They explained the operation of casting out
the 9 s, which was sometimes called the "Hindoo proof." They
contained also the regula falsa and the regula duorum falsorum,
by which algebraical examples could be solved without algebra.
Both these methods were known to the Indians. The regula
falsa or falsa positio was the assigning of an assumed value to
the unknown quantity, which value, if wrong, was corrected
by some process like the "rule of three." Diopliantus used a
method almost identical with this. The regula duorum fal-
sorum was as follows : 7 To solve an equation. /(a?) = F, assume,
for the moment, two values for x ; namely, x = a and $ = 6.
THE ARABS. Ill
Al KuM, the second astronomer at the observatory of the
emir at Bagdad, was a close student of Archimedes and
Apollonius. He solved the problem, to construct a segment
of a sphere equal in volume to a given segment and having
a curved surface equal in area to that of another given seg
ment. He, Al Sagani, and Al Biruni made a study of the
trisection of angles. Abul Gud, an able geometer, solved the
problem by the intersection of a parabola with an equilateral
hyperbola. p
The Arabs had already discovered the theorem that the
sum of two cubes can never be a cube. Abu Mohammed Al
Hogendi of Chorassan thought he had proved this, but we are
told that the demonstration was defective. Creditable work
in theory of numbers and algebra was done by Fahri des Al
Karhi, who lived at the beginning of the eleventh century.
His treatise on algebra is the greatest algebraic work of the
Arabs. In it he appears as a disciple of Diophantus. He
was the first to operate with higher roots and to solve equa
tions of the form x 2n + ax n = b. For the solution of quadratic
equations he gives both arithmetical and geometric proofs.
He was the first Arabic author to give and prove the theorems
on the summation of the series :
33 + ... + n 3
Al Karhi also busied himself with indeterminate analysis.
He showed skill in handling the methods of Diophantus, but
added nothing whatever to the stock of knowledge already
on hand. As a subject for original research, indeterminate
analysis was too subtle for even the most gifted of Arabian
minds. Bather surprising is the fact that Al Karhi s algebra
shows no traces whatever of Hindoo indeterminate analysis.
112 A HISTORY OF MATHEMATICS.
But most astonishing it is, that an arithmetic by the same
author completely excludes the Hindoo numerals. It is con
structed wholly after Greek pattern. Abul Wefa also, in the
second half of the tenth century, wrote an arithmetic in which
Hindoo numerals find no place. This practice is the very
opposite to that of other Arabian authors. The question,
why the Hindoo numerals were ignored by so eminent authors,
is certainly a puzzle. Cantor suggests that at one time there
may have been rival schools, of which one followed almost
exclusively Greek mathematics, the other Indian.
The Arabs were familiar with geometric solutions of quad
ratic equations. Attempts were now made to solve cubic
equations geometrically. They were led to such solutions by
the study of questions like the Archimedean problem, demand
ing the section of a sphere by a plane so that the two seg
ments shall be in a prescribed ratio. The first to state this
problem in form of a cubic equation was Al Mahani of Bagdad,
while Abu Gafar Al Hazin was the first Arab to solve the
equation by conic sections. Solutions were given also by
Al Kuhi, Al Hasan ben Al Haitam, and others. 20 Another
difficult problem, to determine the side of a regular hepta
gon, required the construction of the side from the equation
a 8 _ cc 2 _ 2 x + 1 = 0. It was attempted by many and at last
solved by Abul Cud.
The one who did most to elevate to a method the solution
of algebraic equations by intersecting conies, was Omar al
Hayyami of Chorassan, about 1079 A.D. He divides cubics into
two classes, the trinomial- and quaclrinomial, and each class
into families and species. Each species is treated separately
but according to a general plan. He believed that cubics
could not be solved by calculation, nor bi-quadratics by geom
etry. He rejected negative roots and often failed to discover
all the positive ones. Attempts at bi-quadratic equations
THE ABABS. 113
were made by Abul Wef a, 20 who solved geometrically # 4 = a
land x 4 + <%%? = &
The solution of cubic equations by intersecting conies was
the greatest achievement of the Arabs in algebra. The foun
dation to this work had been laid by the Greeks, for it was
Mensechmus who first constructed the roots of cc 3 a = or
or 3 2 a 3 = 0. It was not his aim to find the number corre
sponding to x, but simply to determine the side a; of a cube
double another cube of side a. The Arabs, on the other
hand, had another object in view : to find the roots of given
numerical equations. In the Occident, the Arabic solutions
of cubics remained unknown until quite recently. Descartes
and Thomas Baker invented these constructions anew. The
works of Al Hayyami, Al Karhi, Abul Gud, show how the
Arabs departed further and further -from the Indian methods,
and placed themselves more immediately under Greek influ
ences. In this way they barred the road of progress against
themselves. The Greeks had advanced to a point where
material progress became difficult with their methods ; but the
Hindoos furnished new ideas, many of which the Arabs now
rejected.
With Al Karhi and Omar Al Hayyami, mathematics among
the Arabs of the East reached flood-mark, and now it begins to
ebb. Between 1100 and 1300 A.D. come the crusades with
war and bloodshed, during which European Christians profited
much by their contact with Arabian culture, then far superior
to their own ; but the Arabs got no science from the Christians
in return. The crusaders were not the only adversaries of the
Arabs. During the first half of the thirteenth century, they
had to encounter the wild Mongolian hordes, and, in 1256, were
conquered by them under the leadership of Hulagu. The
caliphate at Bagdad now ceased to exist. At the close of the
fourteenth century still another empire was formed by Timur
114 A HISTORY OF MATHEMATICS.
or Tamerlane, the Tartar. During such sweeping turmoil, it
is not surprising that science declined. Indeed, it is a marvel
that it existed at all. During the supremacy of Hulagu, lived
Nasir Eddin (1201-1274), a man of broad culture and an able
astronomer. He persuaded Htilagu to build him and his asso
ciates a large observatory at Maraga. , Treatises on algebra,
geometry, arithmetic, and a translation of Euclid s Elements,
were prepared by him. Even at the court of Tamerlane in
Samarkand, the sciences were by no means neglected. A
group of astronomers was drawn to this court. Ulug Beg
(1393-1449), a grandson of Tamerlane, was himself an
astronomer. Most prominent at this time was Al Kaschi, the
author of an arithmetic. Thus, during intervals of peace,
science continued to be cultivated in the- East for several
centuries. The last Oriental writer was Bella- Eddin (1547-
1622). His Essence of Arithmetic stands 011 about the same
level as the work of Mohammed ben Musa Hovarezmi, written
nearly 800 years before.
"Wonderful is the expansive power of Oriental peoples,
with which upon the wings of the wind they conquer half
the world, but more wonderful the energy with which, in
less than two generations, they raise themselves from the
lowest stages of cultivation to scientific efforts." During
all these centuries, astronomy and mathematics in the Orient
greatly excel these sciences in the Occident
Thus far we have spoken only of the Arabs in the East.
Between the Arabs of the East and of the West, which were
under separate governments, there generally existed consider
able political animosity. In consequence of this, and of the
enormous distance between the two great centres of learning,
Bagdad and Cordova, there was less scientific intercourse
among them than might be expected to exist between peoples
having the same religion and written language, Thus the
THE ARABS. 115
course of science in Spain was quite independent of that in
Persia. While wending our way westward to Cordova, we
must stop in Egypt long enough to observe that there, too,
scientific activity was rekindled. ISTot Alexandria, but Cairo
with its library and observatory, was now the home of learn
ing. Foremost among her scientists ranked Ben Junus (died
1008), a contemporary of Abul Wefa. He solved some difficult
problems in spherical trigonometry. Another Egyptian astron
omer was Ibn Al Haitam (died 1038), who wrote on geometric
loci. Travelling westward, we meet in Morocco Abul Hasan
All, whose treatise on astronomical instruments discloses a
thorough knowledge of the Conies of Apollonius. Arriving
finally in Spain at the capital, Cordova, we are struck by the
magnificent splendour of her architecture* At this renowned
seat of learning, schools and libraries were founded during the
tenth century.
Little is known of the progress of mathematics in Spain.
The earliest name that has come down to us is Al Madshriti
(died 1007), the author of a mystic paper on amicable num
bers. 5 His pupils founded schools at Cordova, Dania, and
Granada. But the only great astronomer among the Saracens
in Spain is Gabir ben Aflah of Sevilla, frequently called Geber.
He lived in the second half of the eleventh century. It was
formerly believed that he was the inventor of algebra, and that
the word algebra came from Gabir or Geber. He ranks
among the most eminent astronomers of this time, but, like so
many of his contemporaries, Ms writings contain a great deal
of mysticism. His chief work is an astronomy in nine books, of
which the first is devoted to trigonometry. In his treatment
of spherical trigonometry, he exercises great independence of
thought. He makes war against the time-honoured procedure
adopted by Ptolemy of applying "the rule of six quantities,"
and gives a new way of his own, based on the rule of four
116 A HISTOKY OF MATHEMATICS.
quantities/ This is : If PP : and QQi be two arcs of great
circles intersecting in A, and if PQ and P^ be arcs of great
circles drawn perpendicular to QQ^ then we have the propor
tion
: sin PQ = sin APi : sin
iFrom this he derives the formulas for spherical right triangles.
To the four fundamental formulas already given by Ptolemy,
he added a fifth, discovered by himself. If a, b, c, be the sides,
and A, JB, 0, the angles of a spherical triangle, right-angled at
.4,. then cos B = cos b sin 0. This is frequently called " Geber s
Theorem. 7 Eadical and bold as were his innovations in
spherical trigonometry, in plane trigonometry he followed
slavishly the old beaten path of the Greeks. Not even did he
adopt the Indian sine and cosine/ but still used the Greek
chord of double the angle. So painful was the departure
from old ideas, even to an independent Arab ! After the time
of Gabir ben Aflah there was no mathematician among the
Spanish Saracens of any reputation. In the year in which
Columbus discovered America, the Moors lost their last foot
hold on Spanish, soil.
We have witnessed a laudable intellectual activity among
the Arabs. They had the good fortune to possess rulers
who, by their munificence, furthered scientific research. At
the courts of the caliphs, scientists were supplied with libra
ries and observatories. A large number of astronomical and
mathematical works were written by Arabic authors. Yet
we fail to find a single important principle in mathematics
brought forth by the Arabic mind. Whatever discoveries
they made, were iix fields previously traversed by the Greeks
or the Indians, and consisted of objects which tho latter had
overlooked in their rapid march. The Arabic mind did not
possess that penetrative insight and invention by which mathe
maticians in Europe afterwards revolutionised the science.
EUROPE DURING THE MIDDLE AGES. 117
The Arabs were learned, but not original. Their chief service
to science consists in this, that they adopted the learning of
Greece and India, and kept what they received with scrupu
lous care. When the love for science began to grow in the
Occident, they transmitted to the Europeans the valuable
treasures of antiquity. Thus a Semitic race was, during the
Dark Ages, the custodian of the Aryan intellectual possessions.
EUEOPE DURING THE MIDDLE AGES.
With the third century after Christ begins an era of migra
tion of nations in Europe. The powerful G-oths quit their
swamps and forests in the North and sweep onward in steady
southwestern current, dislodging the Vandals, Sueves, and
Burgundians, crossing the Roman territory, and stopping and
recoiling only when reaching the shores of the Mediterranean.
From the Ural Mountains wild hordes sweep down on the
Danube. The Roman Empire falls to pieces, and the Dark
Ages begin. But dark though they seem, they are the germi
nating season of the institutions and nations of modern Europe.
The Teutonic element, partly pure, partly intermixed with the
Celtic and Latin, produces that strong and luxuriant growth,
the modern civilisation of Europe. Almost all the various
nations of Europe belong to the Aryan stock. As the Greeks
and the Hindoos both Aryan races were the great thinkers
of antiquity, so the nations north of the Alps became the great
intellectual leaders of modern times.
Introduction of Roman Mathematics.
We shall now consider how these as yet barbaric nations of
the North gradually came in possession of the intellectual
118 A HISTOKY OE MATHEMATICS.
treasures of antiquity. Witli the spread of Christianity the
Latin language was introduced not only in ecclesiastical but
also in scientific and all important worldly transactions. Nat
urally the science of the Middle Ages was drawn largely from
Latin sources. In fact, during the earlier of these ages Eo-
man authors were the only ones read in the Occident. Though
Greek was not wholly unknown, yet before the thirteenth
century not a single Greek scientific work had been read or
translated into Latin. Meagre indeed was the science which
could be gotten from Eoman writers, and we must wait several
centuries before any substantial progress is made in mathe
matics.
After the time of Boethins and Cassiodorius mathematical
activity in Italy died out. The first slender blossom of science
among tribes that came from the North was an encyclopaedia
entitled Origines, written by Isidorus (died 636 as bishop of
Seville). This work is modelled after the Eoman encyclopae
dias of Martianus Capella of Carthage and of Cassiodorius.
Part of it is devoted to the quadrivium, arithmetic, music,
geometry, and astronomy. He gives definitions and grammat
ical explications of technical terms, but does not describe the
modes of computation then, in vogue. After Isidorus there
follows a century of darkness which is at last dissipated by
the appearance of Bede the Venerable (672-785), the most
learned man of his time. He was a native of Ireland, then
the home of learning in the Occident. His works contain
treatises on the Computus, or the computation of Easter-time,
and on finger-reckoning. It appears that a finger-symbolism
was then widely used for calculation. The correct determina
tion of the time of Easter was a problem which in those days
greatly agitated the Church. It became desirable to have at
least one monk at each monastery who could determine the
day of religious festivals and could compute the calendar.
EUROPE DUBINa THE MIDDLE AGES. 121
school at Rheims for ten years and became distinguished for
his profound scholarship. By King Otto I. and his successors
Gerbert was held in highest esteem. He was elected bishop
of Rheims, then of Ravenna, and finally was made Pope under
the name of Sylvester II. by his former pupil Emperor Otho
III. He died in 1003, after a life intricately involved in many
political and ecclesiastical quarrels. Such was the career of
the greatest mathematician of the tenth century in Europe.
By his contemporaries his mathematical knowledge was con
sidered wonderful. Many even accused Mm of criminal inter
course with evil spirits.
Gerbert enlarged the stock of his knowledge by procuring
copies of rare books. Thus in Mantua he found the geometr^
of Boethius. Though this is of small scientific value, yet it
is of -great importance in history. It was at that time the
only book from which European scholars could learn the ele
ments of geometry. Gerbert studied it with zeal, and is
generally believed himself to be the author of a geometry.
H. Weissenborn denies his authorship, and claims that the
book in question consists of three parts which cannot come,
from one and the same author. 21 This geometry contains
nothing more than the one of Boethius, but the fact that
occasional errors in the latter are herein corrected shows that
the author had mastered the subject. "The first mathemat
ical paper of the Middle Ages which deserves this name,"
says Hankel, "is a letter of Gerbert to Adalbold, bishop of
Utrecht," in which is explained the reason why the area of a
triangle, obtained " geometrically " by taking the product of
the base by half its altitude, differs from the area calculated
"arithmetically," according to the formula ^a (a + 1), used
by surveyors, where a stands for a side of an equilateral tri
angle. He gives the correct explanation that in the latter
formula all the small squares, in which the triangle is sup-
122 A HISTOEY OF MATHEMATICS.
posed to be divided, are counted in wholly, even though parts
of them project beyond it.
Gerbert made a careful study of the arithmetical works of
Boethius. He himself published two works, Rule of Com
putation on the Abacus, and A Small Book on the Division of
Numbers. They give an insight into the methods of calcu
lation practised in Europe before the introduction of the
Hindoo numerals. Gerbert used the abacus, which was prob
ably unknown to Alcuin. Beraelinus, a pupil of Gerbert,
describes it as consisting of a smooth board upon which geome
tricians were accustomed to strew blue sand, and then to draw
their diagrams. For arithmetical purposes the board was
divided into 30 columns, of which 3 were reserved for frac
tions, while the remaining 27 were divided into groups with
3 columns in each. In every group the columns were marked
respectively by the letters C (centum), I) (decem), and
S (singularis) or M (monas). Bernelinus gives the nine
numerals used, which are the apices of Boethius, and then
remarks that the Greek letters may bo used in their place. 8
By the use of these columns any number can be written
without introducing a zero, and all operations in arithmetic
can be performed in the same way as we execute ours without
the columns, but wiJx the symbol for zero. Indeed, the
methods of adding, subtracting, and multiplying in vogue
among the abacists agree substantially with those of to-day.
But in a division there is very great difference. The early rules
for division appear to have been framed to satisfy the following
three conditions : (1) The use of tho multiplication table shall
be restricted as far as possible; at least, it shall never be
required to multiply mentally a figure of two digits by another
of one digit. (2) Subtractions shall be avoided as much as
possible and replaced by additions. (3) The operation shall
proceed in a purely mechanical way, without requiring trials. 7
EUROPE DUUING THE MIDDLE AG-ES.
123
That it should be necessary to make such conditions seems
strange to us ; but it must be remembered that the monks of
the Middle Ages did not attend school during childhood and
learn xfche multiplication table while the memory was fresh.
Gerbert s rules for division are the oldest extant. They are
so brief as to be very obscure to the uninitiated. They were
probably intended simply to aid the memory by calling to
mind the successive steps in the work. In later manuscripts
they are stated more fully. In dividing any number by another
of one digit ; say 668 by 6, the divisor was first increased to 10
by adding 4. The process is exhibited in the adjoining figure. 8
As it continues, we must imagine the digits
which are crossed out, to be erased and then
replaced by the ones beneath. It is as follows :
600 -*- 10 = 60, but, to rectify the error, 4 x 60,
or 240, must be added ; 200 -*- 10 = 20, but 4 x 20,
or 80, must be added. We now write for
60 + 40 + 80, its sum 180, and continue thus :
100 -T- 10 = 10 ; the correction necessary is 4 x 10,
or 40, which, added to 80, gives 120. Now
100 -*- 10 = 10, and the correction 4 x 10, to
gether with the 20, gives 60. Proceeding as
before, 60 -s- 10 = 6 ; the correction is 4 X^= 24.
Now 20 -5- 10 = 2, the correction being 4x2 = 8.
In the column of units we have now 8 + 4 + 8,
or 20. As before, 20-5-10 = 2; the correction
is 2 x 4 = 8, which, is not divisible by 10, but
only by 6, giving the quotient 1 and the re
mainder 2. All the partial quotients taken
together give 60 + 20 + 10 + 10 +\6+ 2 + 2 + 1 = 111, and
the remainder 2.
Similar but more complicated, is the process when the
divisor contains two or more digits. Were the divisor 27,
124 A HISTOEY OF MATHEMATICS.
tlien tlie next higher multiple of 10, or 30, would be taken
for the divisor, but corrections would be required for the 3.
He who has the patience to carry such a division through
to the end, will understand why it has been said of Gerbert
that "Begulas dedit, quse a sudantibus abacistis vix intelli-
guntur." He will also perceive why the Arabic method of
division, when first introduced, was called the dwisio aurea,
but the one on the abacus, the divisio ferrea.
In his book 011 the abacus, Bernelinus devotes a chapter to
fractions. These are, of course, the duodecimals, first used
by the Eomans. For want of a suitable notation, calculation
with them was exceedingly difficult. It would be so even to
us, were we accustomed, like the early abacists, to express
them, not by a numerator or denominator, but by the appli
cation of names, such as uncia for -^, quincunx for ^, dodrans
for A*
In the tenth century, Gerbert was the central figure among
the learned. In his time the Occident came into secure posses
sion of all mathematical knowledge of the Eomans. During
the eleventh century it was studied assiduously. Though
numerous works were written on arithmetic and geometry,
mathematical knowledge in the Occident was still very insig
nificant. Scanty indeed were the mathematical treasures
obtained from Roman sources.
Translation of Arabic Manuscripts.
By his great erudition and phenomenal activity, Gerbert
infused new life into the study not only of mathematics, but
also of philosophy. Pupils from France, Germany, and Italy
gathered at Eheims to enjoy his instruction. When they
themselves became teachers, they taught of course not only
the use of the abacus and geometry, but also what they had
EUROPE BTTEING THE MIDDLE AGES. 125
learned of the philosophy of Aristotle. His philosophy was
known, at first, only through the writings of Boethius. But
the growing enthusiasm for it created a demand for his com
plete works. Greek texts were wanting. But the Latins
heard that the Arabs, too, were great admirers of Peripatetism,
and that they possessed translations of Aristotle s works and
commentaries thereon. This led them finally to search for
and translate Arabic manuscripts. During this search, mathe
matical works also came to their notice, and were translated
into Latin. Though some few unimportant works may have
been translated earlier, yet the period of greatest activity
began about 1100. The zeal displayed in acquiring the
Mohammedan treasures of knowledge excelled even that of
the Arabs themselves, when, in the eighth century, they
plundered the rich coffers of Greek and Hindoo science.
Among the earliest scholars engaged in translating manu
scripts into Latin was Athelard of Bath. The period of his
activity is the first quarter of the twelfth century. He
travelled extensively in Asia Minor, Egypt, and Spain, and
braved a thousand perils, that he might acquire the language
and science of the Mohammedans. He made the earliest
translations, from the Arabic, of Euclid s Elements and of
the astronomical tables of Mohammed ben Musa Hovarezmi.
In 1857, a manuscript was found in the library at Cambridge,
which proved to be the arithmetic by Mohammed ben Musa
in Latin. This translation also is very probably due to
Athelard.
At about the same time flourished Plato of Tivoli or Plato
Tiburtinus. He effected a translation of the astronomy of
Al Battani and of the SpJicerica of Theodosius. Through the
former, the term sinus was introduced into trigonometry.
About the middle of the twelfth century there was a group
of Christian scholars busily at work at Toledo, under the
126 A HISTORY OF MATHEMATICS.
leadership of Raymond, then archbishop of Toledo, Among
those who worked under his direction, John of Seville was
most prominent. He translated works chiefly on Aristotelian
philosophy. Of importance to us is a liber algorLwii, com
piled by him from Arabic authors. On comparing works like
this with those of the abacists, we notice at once the most
striking difference, which shows that the two parties drew
from, independent sources. It is argued by some that Ger-
bert got his apices and his arithmetical knowledge, not from
BoethiuS; but from the Arabs in Spain, and that part or the
whole of the geometry of Boethius is a forgery, dating from
the time of Gerbert. If this were the case, then the writings
of Gerbert would betray Arabic sources, as do those of John
of Seville. But no points of resemblance are found. Gerbert
could not have learned from the Arabs the use of the abacus,
because all evidence we have goes to show that they did not
employ it, ISTor is it probable that he borrowed from the
Arabs the apices, because they were never used in Europe
except on the abacus. In illustrating an example in division,
mathematicians of the tenth and eleventh centuries state an
example in Roman numerals, then draw an abacus and insert
in it the necessary numbers with the apices. Hence it seems
probable that the abacus and apices were borrowed from the
same source. The contrast between authors like John of
Seville, drawing from Arabic works, and the abacists, consists
in this, that, unlike the latter, the former mention the Hin
doos, use the term algorism, calculate with the zero, and do
not employ the abacus. The former teach the extraction of
roots, the abacists do not; they teach the sexagesimal frac
tions used by the Arabs, while the abacists employ the duo
decimals of the Romans. 8
A little later than John of Seville flourished Gerard of
Cremona in Lombardy. Being desirous to gain possession of
ETJKOPE DURING- THE MIDDLE AGES. 127
the Almagest, he went to Toledo, and there, in 1175, translated
this great work of Ptolemy. Inspired by the richness of
Mohammedan literature, he gave himself up to its study. He
translated into Latin over 70 Arabic works. Of mathematical
treatises, there were among these, besides the Almagest, the
15 books of Euclid, the Sphcerica of Theodosius, a work of
Menelaus, the algebra of Mohammed ben Musa Hovarezmi, the
astronomy of Dshabir ben AfLah, and others less important*
In the thirteenth century, the zeal for the acquisition of
Arabic learning continued. Foremost among the patrons of
science at this time ranked Emperor Frederick II. of Hohen-
staufen (died 1250). Through frequent contact with Mo
hammedan scholars, he became familiar with Arabic science.
He employed a number of scholars in translating Arabic
manuscripts, and it was through him that we came in posses
sion of a new translation of the Almagest. Another royal
head deserving mention as a zealous promoter of Arabic
science was Alfonso X. of Castile (died 1284). He gathered
around Mm a number of Jewish and Christian scholars, who
translated and compiled astronomical works from Arabic
sources. Rabbi Zag and lehuda ben Mose Cohen were the
most prominent among them. Astronomical tables prepared
by these two Jews spread rapidly in the Occident, and con
stituted the basis of all astronomical calculation till the
sixteenth century. 7 The number of scholars who aided in
transplanting Arabic science upon Christian soil was large.
But we mention only one mbre. Giovanni Campano of Novara
(about 1260) brought out a new translation of Euclid, which
drove the earlier ones from the field, and which formed the
basis of the printed editions. 7
At the close of the twelfth century, the Occident was in
possession of the so-called Arabic notation. The Hindoo
methods of calculation began to supersede the cumbrous meth-
128 A HISTORY OF MATHEMATICS,
ods inherited from Borne. Algebra, with, its rules for solving
linear and quadratic equations, had been made accessible to
the Latins. The geometry of Euclid, the Sphwrica of Theodo-
sius ; the astronomy of Ptolemy, and other works were now
accessible in the Latin tongue. Thus a great amount of new
scientific material had come into the hands of the Christians.
The talent necessary to digest this heterogeneous mass of
knowledge was not wanting. The figure of Leonardo of Pisa
adorns the vestibule of the thirteenth century.
It is important to notice that no work either on mathematics
or astronomy was translated directly from the Greek previous
to the fifteenth century.
The First Awakening and its Sequel*
Thus far, France and the British Isles have been the head
quarters of mathematics in Christian Europe. But at the
beginning of the thirteenth century the talent and activity
of one man was sufficient to assign the mathematical science
a new home in Italy. This man was not a monk, like Bede,
Alcuin, or Gerbert, but a merchant, who in the midst of
business pursuits found time for scientific study. Leonardo
of Pisa is the man to whom we owe the first renaissance of
mathematics on Christian soil. He is also called Fibonacci,
i.e. son of Bonaccio. His father was secretary at one of the
numerous factories erected on the south and east coast of the
Mediterranean, by the enterprising merchants of Pisa. He
made Leonardo, when a boy, learn the use of the abacus. The
boy acquired a strong taste for mathematics, and, in later years,
during his extensive business travels in Egypt, Syria, Greece,
and Sicily, collected from the various peoples all the knowl
edge he could get on this subject. Of all the methods of
calculation, he found the Hindoo to be unquestionably the
EUROPE DURING THE MIDDLE AGES. 129
best. Eeturning to Pisa, he published, in 1202, his great
work, the Liber Abaci. A revised edition of this appeared in
1228. This work contains about all the knowledge the Arabs
possessed in arithmetic and algebra, and treats the subject in
a free and independent way. This, together with the other
books of Leonardo, shows that he was not merely a compiler,
or, like other writers of the Middle Ages, a slavish imitator
of the form in which the subject had been previously pre
sented, but that he was an original worker of exceptional
power.
He was the first great mathematician to advocate the adop
tion of the " Arabic notation." The calculation with the zero
was the portion of Arabic mathematics earliest adopted by
the Christians. The nilnds of men had been prepared for the
reception of this by the use of the abacus and the apices.
The reckoning with columns was gradually abandoned, and
the very word abacus changed its meaning and became a
synonym for algorism. For the zero, the Latins adopted
the name zepliirum, from the Arabic sifr (sifra = empty );
hence our English word cipher. The new notation was
accepted readily by the enlightened masses, but, at first,
rejected by the learned circles. The merchants of Italy used
it as early as the thirteenth century, while the monks in the
monasteries adhered to the old forms. In 1299, nearly 100
years after the publication of Leonardo s Liber Abaci, the
Florentine merchants were forbidden the use of the Arabic
numerals in book-keeping, and ordered either to employ the
Roman numerals or to write the numeral adjectives out in
full. In the fifteenth century the abacus with its counters
ceased to be used in Spain and Italy. In France it was used
later, and it did not disappear in England and Germany before
the middle of the seventeenth century. 22 Thus, in the Winter s
Tale (iv. 3), Shakespeare lets the clown be embarrassed by
130 A HISTOEY OF MATHEMATICS.
a problem which, lie could not do without counters. lago
(in Othello, i. 1) expresses his contempt for Michael Casso,
"forsooth, a great mathematician," by calling him a "counter-
caster." So general, indeed, says Peacock, appears to have
been the practice of this species of arithmetic, that its rules
and principles form an essential part of the arithmetical
treatises of that day. The real fact seems to be that the old
methods were used long after the Hindoo numerals were in
common and general use. With such dogged persistency does
man cling to the old !
The Liber Abaci was, for centuries, the storehouse from
which authors got material for works on arithmetic and
algebra. In it are set forth the most perfect methods of
calculation with integers and fractions, known at that time;
the square and cube root are explained ; equations of the first
and second degree leading to problems, either determinate
or indeterminate, are solved by the methods of c single or
double position/ and also by real algebra. The book con
tains a large number of problems. The following was pro
posed to Leonardo of Pisa by a magister in Constantinople,
as a difficult problem : If A gets from B 7 denare, then A s
sum is five-fold B s ; if B gets from A 5 denare, then B 7 s sum
is seven-fold A s. How much has each ? The Liber Abaci
contains another problem, which is of historical interest,
because it was given with some variations by Ahmes, 3000
years earlier : 7 old women go to Home ; each woman has
7 mules, each mule carries 7 sacks, each sack contains 7 loaves,
with each loaf are 7 knives, each knife is put up in 7 sheaths.
What is the sum total of all named? Ans. 137,256. 8
In 1220, Leonardo of Pisa published his l^ractica Geometries,
which contains all the knowledge of geometry and trigonom
etry transmitted to him. The writings of Euclid and of some
other Greek masters were known to him, either from Arabic
ETJBOPE DURING THI MIDDLE AGES. 133
physics and theology. Frivol6us questions, such as "How
many angels can stand on the point of a needle?" were dis
cussed with great interest. Indistinctness and confusion of
ideas characterised the reasoning during this period. Among
the mathematical productions of the Middle Ages, the works
of Leonardo of Pisa appear to us like jewels among quarry-
rubbish. The writers on mathematics during this period were
not few in number, but their scientific efforts were vitiated
by the method of scholastic thinking. Though they possessed
the Elements of Euclid, yet the true nature of a mathematical
proof was so little understood, that Hankel believes it no
exaggeration to say that " since Fibonacci, not a single proof,
not borrowed from Euclid, can be found in the whole literature
of these ages, which fulfils all necessary conditions."
The only noticeable advance is a simplification of numerical
operations and a more extended application of them. Among
the Italians are evidences of an early maturity of arithmetic.
Peacock 22 says : The Tuscans generally, and the Florentines
in particular, whose city was the cradle of the literature and
arts of the thirteenth and fourteenth centuries, were celebrated
for their knowledge of arithmetic and book-keeping, which
were so necessary for their extensive commerce ; the Italians
were in familiar possession of commercial arithmetic long
before the other nations of Europe ; to them we are indebted
for the formal introduction into books of arithmetic, under
distinct heads, of questions in the single and double rule of
three, loss and gain, fellowship, exchange, simple and com
pound interest, discount, and so on.
There was also a slow improvement in the algebraic nota
tion. The Hindoo algebra possessed a tolerable symbolic
notation, which was, however, completely ignored by the Mo
hammedans. In this respect, Arabic algebra approached
much more closely to that of Diophantus, which can scarcely
134 A HISTORY OF MATHEMATICS.
be said to employ symbols in a systematic way. Leonardo of
Pisa possessed no algebraic symbolism. Like the Arabs, lie
expressed the relations of magnitudes to each other by lines
or in words. But in the mathematical writings of the monk
Luca Pacioli (also called Lucas de Eurgo sepulchri) symbols
began to appear. They consisted merely in abbreviations of
Italian words, such as p for piu (more), m for meno (less-), co
for cosa (the thing or unknown quantity). "Our present
notation has arisen by almost insensible degrees as conven
ience suggested different marks of abbreviation to different
authors ; and that perfect symbolic language which addresses
itself solely to the eye, and enables us to take in at a glance
the most complicated relations of quantity, is the result of a
small series of small improvements." ^
We shall now mention a few authors who lived during the
thirteenth and fourteenth and the first half of the fifteenth
centuries. About the time of Leonardo of Pisa (1200 A.D.),
lived the German monk Jordanus Wemorarius, who wrote a once
famous work on the properties of numbers (1496), modelled
after the arithmetic of Boethius. The most trifling numeral
properties are treated with nauseating pedantry and prolixity.
A practical arithmetic based on the Hindoo notation was
also written by him. John Halifax (Sacro Boseo, died 1256)
taught in Paris and made an extract from the Almagest con
taining only the most elementary parts of that work. This
extract was for nearly 400 years a work of great popularity
and standard authority. Other prominent writers are Albertus
Magnus and George Purbach in Germany, and Roger Bacon in
England. It appears that here and there some of our modern
ideas were anticipated by writers of the Middle Ages. Thus,
Nicole Oresme, a bishop in Normandy (died 1382), first con
ceived a notation of fractional powers, afterwards re-dis
covered by Stevinus, and gave rules for operating with them.
EUROPE DURING THE MIDDLE AGES. 135
His notation was totally different from ours. Thomas Brad-
wardine, archbishop of Canterbury, studied star-polygons, a
subject which has recently received renewed attention. The
first appearance of such polygons was with Pythagoras and
his school. We next meet with such polygons in the geom
etry of Boethius and also in the translation of Euclid from
the "Arabic by Athelard of Bath. Bradwardine s philosophic
writings contain discussions on the infinite and the infini
tesimal subjects never since lost sight of. To England
falls the honour of having produced the earliest European
writers on trigonometry. The writings of Bradwardine, of
Bichard of Wallingford, and John Maudith, both professors
at Oxford, and of Simon Bredon of Wincheeombe, contain
trigonometry drawn from Arabic sources.
The works of the Greek monk Maximus Planudes, who lived
in the first half of the fourteenth century, are of interest only
as showing that the Hindoo numerals were then known in
Greece. A writer belonging, like Planudes, to the Byzantine
school, was Moschopulus, who lived in Constantinople in the
early part of the fifteenth century. To him appears to be
due the introduction into Europe of magic squares. He wrote
a treatise on this subject. Magic squares were known to the
Arabs, and perhaps to the Hindoos. Mediaeval astrologers
and physicians believed them to possess mystical properties
and to be a charm against plague, when engraved on silver
plate.
In 1494 was printed the Summa, de Arithmetica, Gfeometria,
Proportione et Proportionalita, written by the Tuscan monk
Lucas Pacioli, who, as we remarked, first introduced symbols
in algebra. This contains all the knowledge of his day on
arithmetic, algebra, and trigonometry, and is the first com
prehensive work which appeared after the Liber Abaci of
Fibonacci. It contains little of importance which cannot be
136 A HISTOBY OF MATHEMATICS.
found in Fibonacci s great work, published three centuries
earlier. 1
Perhaps the greatest result of the influx of Arabic learn
ing was the establishment of universities. What was their
attitude toward mathematics ? The University of Paris, so
famous at the beginning of the twelfth century under the
teachings of Abelard, paid but little attention to this science
during the Middle Ages. Geometry was neglected, and Aris
totle s logic was the favourite study. In 1336, a rule was
introduced that no student should take a degree without
attending lectures on mathematics, and from a commentary
on the first six books of Euclid, dated 1536, it appears that
candidates for the degree of A.M. had to give an oath that
they had attended lectures on these books. 7 Examinations,
when held at all, probably did not extend beyond the first
book, as is shown by the nickname "magister matheseos,"
applied to the Theorem of Pythagoras, the last in the first
book. More^ attention was paid to mathematics at the Univer
sity of Prague, founded 1384. For the Baccalaureate degree,
students were required to take lectures on Sacro Boseo s
famous work on astronomy. Of candidates for the A.M. were
required not only the six books of Euclid, but an additional
knowledge of applied mathematics. Lectures were given on
the Almagest. At the University of Leipzig, the daughter of
Prague, and at Cologne^ less work was required, and, as late
as the sixteenth century, the same requirements were made at
these as at Prague in the fourteenth. The universities of
Bologna, Padua, Pisa, occupied similar positions to the ones
in Germany, only that purely astrological lectures were given
in place of lectures on the Almagest At Oxford, in the
middle of the fifteenth century, the first two books of Euclid
were read. 6
Thus it will be seen that the study of mathematics was
EUROPE DURING THE MIDDLE AGES. 137
maintained at the universities only in a half-hearted manner.
!N"o great mathematician and teacher appeared; to inspire the
students. The best energies of the schoolmen were expended
upon the stupid subtleties of their philosophy. The genius
of Leonardo of Pisa left no permanent impress upon the age,
and another ^Renaissance of mathematics was wanted.
MODERN EUKOPE.
WE find it convenient to choose the time of the capture of
Constantinople by the Turks as the date at which the Middle
Ages ended and Modern Times began. In 1453, the Turks
battered the walls of this celebrated metropolis with cannon,
and finally captured the city $ the Byzantine Empire fell, to
rise no more. Calamitous as was this event to the East, it
acted favourably upon the progress of learning in the West.
A great number of learned Greeks fled into Italy, bringing
with them precious manuscripts of Greek literature. This
contributed vastly to the reviving of classic learning. Up
to this time, Greek masters were known only through the
often very corrupt Arabic manuscripts, but now they began
to be studied from original sources and in their own language.
The first English translation of Euclid was made in 1570 from
the Greek by Sir Henry Billing sley, assisted by John Dee. 29
About the middle of the fifteenth century, printing was in
vented ; books became cheap and plentiful ; the printing-press
transformed Europe into an audience-room. Near the close of
the fifteenth century, America was discovered, and, soon after,
the earth was circumnavigated. The pulse and pace of the
wrld began to quicken. Men s minds became less servile;
they became clearer and stronger. The indistinctness of
thought, which was the characteristic feature of mediaeval
learning, began to be remedied chiefly by the steady cultiva-
138
THE RENAISSANCE. 189
tion of Pure Mathematics and Astronomy. Dogmatism was
attacked; there arose a long struggle with the authority of
the Church and the established schools of philosophy. The
Copernican System was set up in opposition to the time-hon
oured Ptolemaic System. The long and eager contest between
the two culminated in a crisis at the time of Galileo, and
resulted in the victory of the new system. Thus, by slow
degrees, the minds of men were cut adrift from their old
scholastic moorings and sent forth on the wide sea of scientific
inquiry, to discover new islands and continents of truth.
THE RENAISSANCE.
With the sixteenth century began a period of increased
intellectual activity. The human mind made a vast effort to
achieve its freedom. Attempts at its emancipation from
Church authority had been made before, but they were stifled
and rendered abortive. The first great " and successful revolt
against ecclesiastical authority was made in Germany. The
new desire for judging freely and independently in matters
of religion was preceded and accompanied by a growing spirit
of scientific inquiry. Thus it was that, for a time, Germany
led the van in science. She produced Itegiomontanus, Coper
nicus, JRhceticus, ITepler, and Tyclio Brake, at a period when
Prance and England had, as yet, brought forth hardly any
great scientific thinkers. This remarkable scientific produc
tiveness was no doubt due, to a great extent, to the commer
cial prosperity of Germany. Material prosperity is an essential
condition for the progress of knowledge. As long as every
individual is obliged to collect the necessaries for his subsist
ence, there can be no leisure for higher pursuits. At this
time, Germany had accumulated considerable wealth. The
140 A HISTOKY OF MATHEMATICS.
Hanseatic League commanded the trade of the IsTorth. Close
commercial relations existed between Germany and Italy.
Italy, too, excelled in commercial activity and enterprise.
"We need only mention Venice, whose glory began with the cru
sades, and Florence, with her bankers and her manufacturers
of silk and wool. These two cities became great intellectual
centres. Thus, Italy, too, produced men in art, literature, and
science, who shone forth in fullest splendour. In fact, Italy
was the fatherland of what is termed the Eenaissance.
For the first great contributions to the mathematical sciences
we must, therefore, look to Italy and Germany. In Italy
brilliant accessions were made to algebr a, in Germany to
astronomy and trigonometry.
On the threshold of this new era we meet in Germany with
the figure of John Mueller, more generally called Regiomon-
tanus (1436-1476). Chiefly to him we owe the revival of
trigonometry. He studied astronomy and trigonometry at
Vienna under the celebrated George Purbach. The latter
perceived that the existing Latin translations of the Almagest
were full of errors, and that Arabic authors had not remained
true to the Greek original. Purbach therefore began to make
a translation directly from the Greek. But he did not live to
finish it. His work was continued by Eegiomontanus, who
went beyond his master. Eegiomontanus learned the Greek
language from Cardinal Bessarion, whom he followed to Italy,
where he remained eight years collecting manuscripts from
Greeks who had fled thither from the Turks. In addition to
the translation of and the commentary on the Almagest, he
prepared translations of the Conies of Apollonius, of Archi
medes, and of the mechanical works of Heron. Eegiomontanus
and Purbach adopted the Hindoo sine in place of the Greek
chord of double the arc. The Greeks and afterwards the Arabs
divided the radius into 60 equal parts, and each of these again
THE RENAISSANCE. 141
into 60 smaller ones. The Hindoos expressed the length of
the radius by parts of the circumference, saying that of the
21,600 equal divisions of the latter, it took 3438 to measure
the radius. Begioniontanus, to secure greater precision, con
structed one table of sines on a radius divided into 600,000
parts, and another on a radius divided decimally into 10,000,000
divisions. He emphasised the use of the tangent in trigonom
etry. Following out some ideas of his master, he calculated
a table of tangents. German mathematicians were not the
first Europeans to use this function. In England it was known
a century earlier to Bradwardine, who speaks of tangent (umbra
recta) and cotangent (umbra versa), and to John Maudith.
Begiomontanus was the author of an arithmetic and also of
a complete treatise on trigonometry, containing solutions of
both plane and spherical triangles. The form which he gave
to trigonometry has been retained, in its main features, to the
present day.
Begiomontanus ranks among the greatest men that Germany
has ever produced. His complete mastery of astronomy and
mathematics, and his enthusiasm for them, were of far-
reaching influence throughout Germany. So great was his
reputation, that Pope Sixtus IV. called him to Italy to
improve the calendar. Begiomontanus left his beloved city
of JSTurnberg for Borne, where he died in the following year.
After the time of Purbach and Begiomontanus, trigonome
try and especially the calculation of tables continued to occupy
German scholars. More refined astronomical instruments were
made, which gave observations of greater precision ; but these
would have been useless without trigonometrical tables of cor
responding accuracy. Of the several tables calculated, that
by Georg Joachim of Feldkirch in Tyrol, generally called
Rhaeticus, deserves special mention. He calculated a table of
sines with the radius =10,000,000,000 and from 10" to 10";
142 A HISTORY OF MATHEMATICS.
and, later on, another with the radius = 1,000,000,000,000,000,
and proceeding from 10" to 10". He began also the con
struction of tables of tangents and secants, to be carried to
the same degree of accuracy; but he died before finishing them.
For twelve years he had htfd in continual employment several
calculators. The work wa*s completed by his pupil, Valentine
Otho, in 1596. This was: indeed a gigantic work, a monu
ment of German diligence and indefatigable perseverance.
The tables were republished in 1613 by Pitiscus, who spared
no pains to free them of errors. Astronomical tables of
so great a degree of accuracy had never been dreamed of
by the Greeks, Hindoos, or Arabs. That Ehseticus was not a
ready calculator only, is indicated by his views on trignoraet-
rical lines. Up to his time, the trigonometric functions had
been considered always with relation to the arc ; he was the
first to construct the right triangle and to make them depend
directly upon its angles. It was from the right triangle that
Ehseticus go this idea of calculating the hypotenuse ; i.e. he
was the first to plan a table of secants. Good work in trigo
nometry was done also by Vieta and Komanus.
We shall now leave the subject of trigonometry to witness
the progress in the solution of algebraical equations. To do
so, we must quit Germany for Italy. The first comprehensive
algebra printed was that of Lucas Pacioli. He closes his
book by saying that the solution of the equations o> 3 + mx = n,
x? + n = mx is as impossible at the present state of science as
the quadrature of the circle. This remark doubtless stimu
lated thought. The first step in the algebraic solution of
cubics was taken by Scipio Ferro (died 1526), a professor of
mathematics at Bologna, who solved the equation o? + mx = n.
Nothing more is known of his discovery than that he imparted
it to his pupil, Floridas, in 1505. It was the practice in those
days and for two centuries afterwards to keep discoveries
THE RENAISSANCE. 143
secret, in order to secure by that means an advantage over
rivals by proposing problems beyond their reach. This prac
tice gave rise to numberless disputes regarding the priority of
inventions. A second solution of eubics was given by Nicolo
of Brescia (1506(?)-1557). When a boy of six, Nicolo was
so badly cut by a French soldier that he never again gained
the free use of his tongue. Hence he was called Tartaglia,
i.e. the stammerer. His widowed mother being too poor to
pay his tuition in school, he learned to read and picked up a
knowledge of Latin, Greek, and mathematics by himself.
Possessing a mind of extraordinary power, he was able to
appear as teacher of mathematics at an early age. In 1530,
one Colla proposed him several problems, one leading to the
equation &+px 2 = q. Tartaglia found an imperfect method
for solving this, but kept it secret. He spoke about his secret
in public and thus caused Ferro s pupil, Floridas, to proclaim
his own knowledge of the form a? + mx*=n. Tartaglia, believ
ing him to be a mediocrist and braggart, challenged him to a
public discussion, to take place on the 22d of February, 1535.
Hearing, meanwhile, that his rival had gotten the method
from a deceased master, and fearing that lie would be beaten
in the contest, Tartaglia put in all the zeal, industry, and
skill to find the xule for the equations, and he succeeded in it
ten days before the appointed date, as he himself modestly
says. 7 The most difficult step was, no doubt, the passing from
quadratic irrationals, used in operating from time of old, to
cubic irrationals. Placing # = ^-~^, Tartaglia perceived
that the irrationals disappeared from the equation re 3 4- mx = n,
making n=t u. But this last equality, together with
(-|m) 3 = tu, gives at once
14:4 A H13TOBY OJF MATHEMATICS.
This is Tartaglia s solution of a? 4- mx = n. On the 13th of
February, he found a similar solution for cc 3 = mx + n. The
contest began on the 22d. Each contestant proposed thirty
problems. The one who could solve the greatest number within
fifty days should be the victor. Tartaglia solved the thirty
problems proposed by Floridas in two hours ; Floridas could
not solve any of Tartaglia s. From now on, Tartaglia studied
cubic equations with a will. In 1541 he discovered a general
solution for the cubic cc 3 px 2 = q, by transforming it into
the form a? mx=n. The news of Tartaglia s victory
spread all over Italy. Tartaglia was entreated to make known
his method, but he declined to do so, saying that after his
completion of the translation from the Greek of Euclid and
Archimedes, he would publish a large algebra containing his
method. But a scholar from Milan, named Eieronimo Cardano
(1501-1576), after many solicitations, and after giving the
most solemn and sacred promises of secrecy, succeeded in
obtaining from Tartaglia a knowledge of his rules.
At this time Cardan was writing his Ars Magna, and he
knew no better way to crown his work than by inserting the
much sought for rules for solving cubics. Thus Cardan broke
his most solemn vows, and published in 1545 in his Ars Magna
Tartaglia s solution of cubics. Tartaglia became desperate.
His most cherished hope, of giving to the world an immortal
work which should be the monument of his deep learning and
power for original research, was -suddenly destroyed; for the
crown intended for his work had been snatched, away. His
first step was to write a history of his invention ; but, to com
pletely annihilate his enemies, he challenged Cardan and his
pupil Lodovico Ferrari to a contest : each party should propose
thirty-one questions to be solved by the other within fifteen
days. Tartaglia solved most questions in seven days, but the
other party did not send in their solution before the expiration
THE RENAISSANCE. 145
of the fifth month; moreover, all their solutions except one
were wrong. A replication and a rejoinder followed. Endless
were the problems proposed and solved on both sides. The
dispute produced much chagrin and heart-burnings to the par
ties, and to Tartaglia especially, who met with many other
disappointments. After having recovered himself again, Tar
taglia began, in 1556, the publication of the work which he
had had in his mind for so long; but he died before he reached
the consideration of cubic equations. Thus the fondest wish
of his life remained unfulfilled ; the man to whom we owe the
greatest contribution to algebra made in the sixteenth century
was forgotten, and his method came to be regarded as the dis
covery of Cardan and to be called Cardan s solution.
Remarkable is the great interest that the solution of cubics
excited throughout Italy. It is but natural that after this
great conquest mathematicians should attack bi-quadratic equa
tions. As in the case of cubics, so here, the first impulse was
given by Colla, who, in 1540, proposed for solution the equa
tion #* + 6 v? + 36 = 60 x. To be . sure, Cardan had studied
particular cases as early as 1539. Thus he solved the equation
13 of = x 4 + 2 x* + 2 x + 1 by a process similar to that em
ployed by Diophantus and the Hindoos ; namely, by adding
"to both sides 3 of and thereby rendering both numbers
complete squares. But Cardan failed to find a general solu
tion; it remained for his pupil Ferrari to prop the reputa
tion of his master by the brilliant discovery of the general
solution of bi-quadratic equations. Perrari reduced Colla s
equation to the form (o; 2 -f 6) 2 = 60^ + 6^. In order to
give also the right member the form of a complete square
he added to both members the expression 2 (y? -f 6) y + y 2 ,
containing a new unknown quantity y. This gave him (a? + 6
+ y)* = (6 + 2 y) 01? + 60 x + (12 y + y 2 ) . The condition that
the right member be a complete square is expressed by the
146 A HISTOBY OF MATHEMATICS.
cubic equation (2y + 6) (12 y + y 2 ) = 900. Extracting the
square root of the bi-quadratic, he got x 2 + 6 + y = x V5~y+~6
+ Solving the cubic for y and substituting, it re-
V2 2/4-6
mained only to determine x from the resulting quadratic.
Ferrari pursued a similar method with other numerical bi
quadratic equations. 7 Cardan had the pleasure of publishing
this discovery in his Ars Magna in 1545. Ferrari s solution
is sometimes ascribed to BombelH, but he is no more the dis
coverer of it than Cardan is of the solution called by his
name.
To Cardan algebra is much indebted. In his ATS Magna
he takes notice of negative roots of an equation, calling them
fictitious, while the positive roots are called real. Imaginary
roots he does not consider; cases where they appear he calls
impossible. Cardan also observed the difficulty in the irre
ducible case in the cubics, which, like the quadrature of the
circle, has since " so much tormented the perverse ingenuity of
mathematicians." But he did not understand its nature. It re
mained for Raphael Bombelli of Bologna, who published in 1572
an algebra of great merit, to point out the reality of the appar
ently imaginary expression which the root assumes, and thus
to lay the foundation of a more intimate knowledge of imagi
nary quantities.
After this brilliant success in solving equations of the third
and fourth degrees, there was probably no one who doubted,
that with aid of irrationals of higher degrees, the solution of
equations of any degree whatever could be found. But all
attempts at the algebraic solution of the quintic were fruitless,
and, finally, Abel demonstrated that all hopes of finding alge
braic solutions to equations of higher than the fourth degree
were purely Utopian.
Since no solution by radicals of equations of higher degrees
THE RENAISSANCE. 151
metic of Grammateus, a teacher at the University of Yienna.
His pupil, Christoff Rudolff, the writer of the first text-book
on algebra in the German language (printed in 1525), employs
these symbols also. So did Stifel, who brought out a second
edition of HudolfFs Goss in 1553. Thus, by slow degrees,
their adoption became universal. There is another short-hand
symbol of which we owe the origin to the Germans. In a
manuscript published sometime in the fifteenth century, a dot
placed before a number is made to signify the extraction of a
root of that number. This dot is the embryo of our present
symbol for the square root. Christoff Rudolff, in his algebra,
remarks that " the "radif guadrata is, for brevity, designated
in his algorithm with the character >/, as y^." Here the dot
has grown into a symbol much like our own. This same
symbol was used by Micliael Stifel. Our sign of equality is due
to Robert Recorde (1510-1558), the author of The WJietstone of
Witte (1557), which is the first English treatise on algebra.
He selected this symbol because no two things could be more
equal than two parallel lines =. The sign -* for division was
first used by Johann Heinrich Rahn, a Swiss, in 1659, and
was introduced in England by John Pell in 1668.
Micliael Stifel (1486?-1567), the greatest German algebraist
of the sixteenth century, was born in Esslingen, and died in
Jena. He was educated in the monastery of his native place,
and afterwards became Protestant minister. The study of the
significance of mystic numbers in Eevelation and in Daniel
drew him to mathematics. He studied German and Italian
works, and published in 1544, in Latin, a book entitled
Arithmetica Integra. Melanchthon wrote a preface to it. Its
three parts treat respectively of rational numbers, irrational
numbers, and algebra. Stifel gives a table containing the nu
merical values of the binomial coefficients for powers below the
18th. He observes an advantage in letting a geometric progres-
152 A HISTORY OF MATHEMATICS.
sion correspond to an arithmetical progression, and arrives at
the designation of integral powers by numbers. Here are the
germs of the theory of exponents. In 1545 Stifel published
an arithmetic in German. His edition of Kudolffs Goss con
tains rules for solving cubic equations, derived from the
the writings of Cardan.
We remarked above that Vieta discarded negative roots of
equations. Indeed, we find few algebraists before and during
the Renaissance who understood the significance even of
negative quantities. Fibonacci seldom uses them. Pacioli
states the rule that "minus times minus gives plus," but
applies it really only to the development of the product of
(a &) (c d) ; purely negative quantities do not appear in
his work. The great German "Cossist" (algebraist), Michael
Stifel, speaks as early as 1544 of numbers which are " absurd "
or " fictitious below zero," and which arise when " real numbers
above zero " are subtracted from zero. Cardan, at last, speaks
of a "pure minus " ; "but these ideas," says Hankel, "remained
sparsely, and until the beginning of the seventeenth century,
mathematicians dealt exclusively with absolute positive quan
tities." The first algebraist who occasionally places a purely
negative quantity by itself on one side of an equation, is
Harriot in England. As regards the recognition of negative
roots, Cardan and Bombelli were far in advance of all writers
of the Eenaissance, including Vieta. Yet even they mentioned
these so-called false or fictitious roots only in passing, and
without grasping their real significance and importance. On
this subject Cardan and Bombelli had advanced to about the
same point as had the Hindoo Bhaskara, who saw negative roots,
but did not approve of them. The generalisation of the con
ception of quantity so as to include the negative, was an
exceedingly slow and difficult process in the development of
algebra.
THE RENAISSANCE. 158
We shall now consider the history of geometry during the
Renaissance. Unlike algebra, it made hardly any progress.
The greatest gain was a more intimate knowledge of G-reek
geometry. No essential progress was made before the time of
Descartes. Begiomontanus, Xylander of Augsburg, Tartaglia,
Commandinus of Urbino in Italy, Maurolycus, and others,
made translations of geometrical works from the Greek. John
Werner of Eurnberg published in 1522 the first work on
conies which appeared in Christian Europe. Unlike the
geometers of old, he studied the sections in relation with the
cone, and derived their properties directly from it. This mode
of studying the conies was followed by Maurolyctis of Messina
(1494-1575). The latter is, doubtless, the greatest geometer
of the sixteenth century. Prom the notes of Pappus, he
attempted to restore the missing fifth book of Apollonius
on maxima and minima. His chief work is his masterly
and original treatment of the conic sections, wherein he dis
cusses tangents and asymptotes more fully than Apollonius
had done, and applies them to various physical and astronomi
cal problems.
The foremost geometrician of Portugal was Nonius; of
Prance, before Yieta, was Peter Ramus, who perished in the
massacre of St. Bartholomew. Vieta possessed great famil
iarity with ancient geometry. The new form which he gave
to algebra, by representing general quantities by letters, en
abled him to point out more easily how the construction of
the roots of cubics depended upon the celebrated ancient prob
lems of the duplication of the cube and the trisection of an
angle. He reached the interesting conclusion that the former
problem includes the solutions of all cubics in which the radi
cal in Tartaglia s formula is real, but that the latter problem
includes only those leading to the irreducible case.
The problem of the quadrature of the circle was revived in
154 A HISTORY OF MATHEMATICS.
this age, and was zealously studied even by men of eminence
and mathematical ability. The army of circle-squarers became
most formidable during the seventeenth century. Among the
first to revive this problem was the German Cardinal Mcolaus
Cusanus (died 1464), who had the reputation of being a great
logician. His fallacies were exposed to full view by Eegio-
montanus. As in this case ; so in others, every quadrator of
note raised up an opposing mathematician : Orontius was met
by Buteo and Nonius; Joseph Scaliger by Vieta, Adrianus
Eomanus, and Clavius ; A. Quercu by Peter Metius. Two
mathematicians of Netherlands, Adrianus Romanus and Ludolph
van Ceulen, occupied themselves with approximating to the
ratio between the circumference and the diameter. The for
mer carried the value TT to 15, the latter to 35 ; places. The
value of TT is therefore often named "Ludolph s number." His
performance was considered so extraordinary, that the num
bers were cut on his tomb-stone in St. Peter s church-yard, at
Leyden. Eomanus was the one who prppounded for solution
that equation of the forty-fifth degree solved by Yieta. On
receiving Vieta s solution, he at once departed for Paris, to
make his acquaintance with so great a master, Vieta pro
posed to him the Apollonian problem, to draw a circle touching
three given circles. " Adrianus Eomanus solved the problem
by the intersection of two hyperbolas ; but this solution did not
possess the rigour of the ancient geometry. Yieta caused him
to see this, and then, in his turn, presented a solution which
had all the rigour desirable." 25 Eomanus did much toward
simplifying spherical trigonometry by reducing, by means of
certain projections, the 28 cases in triangles then considered
to only six.
Mention must here be made of the improvements of the
Julian calendar. The yearly determination of the movable
feasts had for a long time been connected with an untold
THE BBKAISSANCB. 155
amount of confusion. The rapid progress of astronomy led
to the consideration of this subject, and many new calendars
were proposed. Pope Gregory XIII. convoked a large number
of mathematicians, astronomers, and prelates, who decided
upon the adoption of the calendar proposed by the Jesuit
Lilius Clavius.* To rectify the errors of the Julian calendar
it was agreed to write in the new calendar the 15th of
October immediately after the 4th of October of the year
1582. The Gregorian calendar met with a great deal of oppo
sition botji among scientists and among Protestants. Clavius,
who ranked high as a geometer, met the objections of the
former most ably and effectively ; the prejudices of the latter
passed away with time.
The passion for the study of mystical properties of numbers
descended from the ancients to the moderns. Much was
written on numerical mysticism even by such eminent men
as Pacioli and Stifel. The Numerorum Hysteria of Peter
Bungus covered 700 quarto pages. He worked with great
industry and satisfaction on 666, which is the number of the
beast in Revelation (xiii. 18), the symbol of Antichrist. He
reduced the name of the ( impious Martin Luther to a form
which may express this formidable number. Placing a = 1,
& = 2, etc., Jc = 10, I = 20, etc., he finds, after misspelling the
name, that M (30 ) A^) K^ T (100 ) I (9 ) Is (40 ) L (20) v (2 oo) T (100 ) E (5) .E (SO ) A (1)
constitutes the number required. These attacks on the great
reformer were not unprovoked, for his friend, Michael Stifel,
the most acute and original of the early mathematicians of
Germany) exercised an equal ingenuity in showing that the
above number referred to Pope Leo X., a demonstration
which gave Stifel unspeakable comfort. 22
Astrology also was still a favourite study. It is well
known that Cardan, Maurolycus, Regiomontanus, and many
other eminent scientists who lived at a period even later than
156 A HISTORY OF MATHEMATICS.
this, engaged in deep astrological study ; but it is not so gen
erally known that besides the occult sciences already named,
men engaged in the mystic study of star-polygons and magic
squares. " The pentagranima gives you pain," says Faust to
Mephistopheles. It is of deep psychological interest to see
scientists, like the great Kepler, demonstrate on one page a
theorem on star-polygons, with strict geometric rigour, while
on the next page, perhaps, he explains their use as amulets
or in conjurations. 1 Playfair, speaking of Cardan as an astrol
oger, calls him "a melancholy proof that there is no folly
or weakness too great to be united to high intellectual attain
ments." 26 Let our judgment not be too harsh. The period
under consideration is too near the Middle Ages to admit of
complete emancipation from mysticism even among scientists.
Scholars like Kepler, Xapier, Albrecht Duerer, while in the
van of progress and planting one foot upon the firm ground
of truly scientific inquiry, were still resting with the other
foot upon the scholastic ideas of preceding ages.
VIETA TO DESCABTES.
The ecclesiastical power, which in the ignorant ages was an
unmixed benefit, in more enlightened ages became a serious
evil. Thus, in France, during the reigns preceding that of
Henry IV., the theological spirit predominated. This is pain
fully shown by the massacres of Vassy and of St. Bartholo
mew. Being engaged in religious disputes, people had no
leisure for science and for secular literature. Hence, down
to the time of Henry IV., the ^rench_llhad n.oi. puF "forth a
single work, the destruction of which^,ould now be a loss to
Europe," In England,,, on the other hand, no religious wars
were waged.. ...The people were comparatively indifferent about
VIETA TO DESCARTES. 157
religious strifes ; they concentrated their ability upon secular
matters, and acquired, in the sixteenth century, a literature
which is immortalised by the genius of Shakespeare and
Spenser* This great literary age in England was followed
by a great scientific age. At the close of the sixteenth cen
tury, the shackles of ecclesiastical authority were thrown off
by France. The ascension of Henry IV. to the throne was
followed in 1598 by the Edict of Nantes, granting freedom
of worship to the Huguenots, and thereby terminating religious
wars. The genius of the French nation now began to blossom.
Cardinal Richelieu, during the reign of Louis XIII., pursued
the broad policy of not favouring the opinions of any sect, but
of promoting the interests of the nation. His age was re
markable for the progress of knowledge. It produced that
great secular literature, the counterpart of which was found
in England in the sixteenth century. The seventeenth cen
tury was made illustrious also by the great French mathema
ticians, Eoberval, Descartes, Desargues, Fermat, and Pascal.
More gloomy is the picture in Germany. The great changes
which revolutionised the world in the sixteenth century, and
which led England to national greatness, led Germany to
degradation. The first effects of the Eeformation there were
salutary. At the close of the fifteenth and during the six
teenth century, Germany had been conspicuous for her scien
tific pursuits. She had been the leader in astronomy and
trigonometry. Algebra also, excepting for the discoveries in
cubic equations, was, before the time of Vieta, in a more
advanced state there than elsewhere. But at the beginning
of the seventeenth century, when the sun of science began to
rise in "France, it- set in Germany. Theologic disputes and
religious strife ensued. The Thirty Years War (1618-1648)
proved ruinous. The German empire was shattered, and
became a mere lax confederation of petty despotisms. Com-
158 A HISTOBY OF MATHEMATICS.
- inerce was destroyed ; national feeling died out. Art disap
peared, and in literature there was only a slavish imitation
of French artificiality. ISTor did Germany recover from this
low state for 200 years ; for in 1756 began another struggle,
the Seven Years War, which tnrned Prussia into a wasted
land. Thus it followed that at the beginning of the seven
teenth century, the great Kepler was the only German mathe
matician of eminence, and that in the interval of 200 years
between Kepler and Gauss, there arose no great mathematician
in Germany excepting Leibniz.
Up to the seventeenth century, mathematics was cultivated
but little in Great Britain. During the sixteenth century, she
brought forth no mathematician comparable with Yieta, Stifel,
or Tartaglia. But with the time of Eecorde, the English
became conspicuous for numerical skill. The first important
irithmetical work of English authorship was published in
Latin in 1522 by Cuthbert Tonstall (1474-1559). He had
studied at Oxford, Cambridge, and Padua, and drew freely
from the works of Pacioli and Eegiomontanus. Eeprints of
his arithmetic appeared in England and Prance. After
Recorde the higher branches of mathematics began to be
studied. Later, Scotland brought forth Napier, the inventor
of logarithms. The instantaneous appreciation of their value
is doubtless the result of superiority in calculation. In Italy,
and especially in France, geometry, which for a long time had
been an almost stationary science, began to be studied with
Success. Galileo, Torricelli, Eoberval, Permat, Desargues,
Pascal, Descartes, and the English Wallis are the great revo-
lutioners of this science. Theoretical mechanics began to be
studied. The foundations were laid by Permat and Pascal
for the theory 0f numbers and the theory of probability.
We shall first consider the improvements made in the art
of calculating. The nations of antiquity experimented thou-
VIETA TO DESCARTES. 159
sands of years upon numeral notations before they happened
to strike upon the so-called " Arabic notation." In the simple
expedient of the cipher, which was introduced by the Hindoos
about the fifth or sixth century after Christ, mathematics re
ceived one of the most powerful impulses. It would seem that
after the "Arabic notation " was once thoroughly understood,
decimal fractions would occur at once as an obvious extension
of it. But "it is curious to think how much science had
attempted in physical research and how deeply numbers had
been pondered, before it was perceived that the all-powerful
simplicity of the Arabic notation 5 was as valuable and as
manageable in an infinitely descending as in an infinitely
ascending progression." ^ Simple as decimal fractions appear
to us, the invention of them is not the result of one mind or
even of one age. They came into use by almost imperceptible
degrees. The first mathematicians identified with their his
tory did not perceive their true nature and importance, and
failed to invent a suitable notation. The idea of decimal
fractions makes its first appearance in methods for approxi
mating to the square roots of numbers. Thus John of Seville,
presumably in imitation of Hindoo rules, adds 2n ciphers
to the number, then finds the square root, and takes this
as the numerator of a fraction whose denominator is 1 fol
lowed by n ciphers. The same method was followed by
Cardan, but it failed to be generally adopted even by his
Italian contemporaries ; for otherwise it would certainly have
been at least mentioned by Oataldi (died 1626) in a work
devoted exclusively to the extraction of roots. Cataldi finds
the square root by means of continued fractions a method
ingenious and novel, but for practical purposes inferior to
Cardan s. Orontius Finaeus (died 1555) in--France, and Wil
liam Buckley (died about 1550) in England extracted the"
square rootfin the same way as Cardan and John of Seville.
160 A HISTORY OF MATHEMATICS.
The invention of decimals is frequently attributed to Regio
montanus, on the ground that instead of placing the sinus
totus7*nT "trigonometry, equal to a multiple of 60, like the
Greeks, he put it = 100,000. But here the trigonometrical
lines were expressed in integers, and not in fractions. Though
he adopted a decimal division of the radius, he and his suc
cessors did not apply the idea outside of trigonometry and,
indeed, had no notion whatever of decimal fractions. To
j$imon_Jtevin of Bruges in Belgium (1548-1620), a man who
did a great deal of work in most diverse fields of science, we
owe the first systematic treatment of decimal fractions. In
his La Disme (1585) he describes in very express terms the
advantages, not only of decimal fractions, but also of the
decimal division in systems of weights and measures. Stevin
applied the new fractions " to all the operations of ordinary
.arithmetic." 25 What he lacked was a suitable notation. In
place of our decimal point, he used a cipher ; to each place in
the fraction was attached the corresponding index. Thus, in
0123
his notation, the number 5.912 would be 5912 or 59(i)l@2.
These indices, though cumbrous in practice, are of interest,
because they are the germ of an important innovation. To
Stevin belongs the honour of inventing our present mode of
designating powers and also of introducing fractional expo
nents into algebra. Strictly speaking, this had been done much
earlier by Oresme, but it remained wholly unnoticed. -Not even
Stevin s innovations were immediately appreciated or at once
accepted, but, unlike Oresme s, they remained a secure posses
sion. $To improvement was made in the notation of decimals
till the beginning of the seventeenth century. After Stevin,
decimals were used by Joost Biirgi, a Swiss by birth, who pre
pared a manuscript on arHlmrotic soon after 1592, and by
Joliann Hartmann Beyer, who assumes the invention as his own.
In 1603, he published at Frankfurt on the Main a Logistica
YIETA TO DESCABTES. 165
Napier and Briggs, Adrian Vlacq of Gouda in Holland. He
published in 1628 a table of logarithms from 1 to 100,000, of
which 70,000 were calculated by himself. The first publication
of Briggian logarithms .of trigonometric functions was made
in 1620 by Gunter, a colleague of Briggs, who found the loga
rithmic sines and tangents for every minute to seven places.
Gunter was the inventor of the words cosine and cotangent.
Briggs devoted the last years of his life to calculating more
extensive Briggian logarithms of trigonometric functions, but
he died in 1631, leaving his work unfinished. It was carried
on by the English Henry Gellibrand, and then published by
Vlacq. at his own expense. Briggs divided a degree into 100
parts, but owing to the publication by Vlacq of trigonometrical
tables constructed on the old sexagesimal division, Briggs
innovation remained unrecognised. Briggs and Vlacq published
four fundamental works, the results of which " have never been
superseded by any subsequent calculations."
The first logarithms upon the natural base e were published
by John Speidell in his New Logarithmes (London, 1619), which
contains the natural logarithms of sines, tangents, arid secants.
The only possible rival of John Napier in the invention of
logarithms was the Swiss Justus Byrgius (Joost Burgi). He
published a rude table of logarithms six years after the
appearance of the Canon Mirificus, but it appears that he
conceived the idea and constructed that table as early, if not
earlier, than Napier did his. But he neglected to have the
results published until Napier s logarithms were known and
admired throughout Europe.
Among the various inventions of Napier to assist the
memory of the student or calculator, is "Napier s rule of
.circular parts" for the solution of spherical right triangles.
It is, perhaps, "the happiest example of artificial memory
that is known."
166 A HISTORY OF MATHEMATICS
The most brilliant conquest in algebra during the sixteenth,
century had been the solution of cubic and bi-quadratic equa
tions. All attempts at solving algebraically equations of higher
degrees remaining fruitless, a new line of inquiry the prop
erties of equations and their roots was gradually opened up.
"We have seen that Yieta had attained a partial knowledge of
the relations between roots and coefficients. Peletarius, a
[Frenchman, had observed as early as 1558, that the root of an
equation is a divisor- of the last term. One who extended the
theory of equations somewhat further than Vieta, was Albert
Girard (1590-1634), a Flemish mathematician. Like Vieta,
this ingenious author applied algebra to geometry, and was
the first who understood the use of negative roots in the
solution of geometric problems. He spoke of imaginary quan
tities ; inferred by induction that every equation has as many
roots as there are units in the number expressing its degree ;
and first showed how to express the sums of their powers in
terms of the coefficients. Another algebraist of considerable
power was the English Thomas Harriot (1560-1621). He
accompanied the first colony sent out by Sir Walter Raleigh
to Yirginia. After having surveyed that country he returned
to England. As a mathematician, he was the boast of his
country. He brought the theory of equations under one
comprehensive point of view by grasping that truth in its
full extent to which Yieta and Girard only approximated ; viz.
that in an equation in its simplest form, the coefficient of
the second term with its sign changed is equal to the sum of
the roots ; the coefficient of the third is equal to the sum
of the products of every two of the roots ; etc. He was the
first to decompose equations into their simple factors ; but,
since he failed to recognise imaginary and even negative roots,
he failed also to prove that every equation could be thus
decomposed. Harriot made some changes in algebraic nota-
VIETA TO DESCARTES. 167
tion, adopting small letters of the alphabet in place of the
capitals used by Vieta. The symbols of inequality > and <
were introduced by him. Harriot s work, Artis Analytical
praxis, was published in 1631, ten years after his death.
William Oughtred (1574-1660) contributed vastly to the propa
gation of mathematical knowledge in England by his treatises,
which were long used in the universities. He introduced x
as symbol of multiplication, and : : as that of proportion. By
him ratio was expressed by only one dot. In the eighteenth
century Christian Wolf secured the general adoption of the
dot as a symbol of multiplication, and the sign for ratio was
thereupon changed to two dots. Oughtred s ministerial duties
left him but little time for the pursuit of mathematics during
daytime, and evenings his economical wife denied him the
use of a light.
Algebra was now in a state of sufficient perfection to enable
Descartes to take that important step which forms one of the
grand epochs in the history of mathematics, the application
of algebraic analysis to define the nature and investigate the
properties of algebraic curves.
In geometry, the determination of the areas of curvilinear
figures was diligently studied at this period. Paul Guldin
(1577-1643), a Swiss mathematician of considerable note,
rediscovered the following theorem, published in his Centro-
baryca, which has been named after him, though first found
in the Mathematical Collections of Pappus : The volume of a
solid of revolution is equal to the area -of the generating
figure, multiplied by the circumference described by the centre
of gravity. We shall see that this method excels that of
Kepler and Cavalieri in following a more exact and natural
course ; but it has the disadvantage of necessitating the deter
mination of the centre of gravity, which in itself may be a
more difficult problem than the original one of finding the
168 A HISTORY OF MATHEMATICS.
volume. Guldin made some attempts to prove his theorem,
but Cavalieri pointed out the weakness of his demonstration.
Johannes Kepler (1571-1630) was a native of Wurtemberg
and imbibed Copernican principles while at the University of
Tubingen. His pursuit of science was repeatedly interrupted
by war, religious persecution, pecuniary embarrassments, fre
quent changes of residence, and family troubles. In 1600 he
became for one year assistant to the Danish astronomer, Tycho
Brahe, in the observatory near Prague. The relation between
the two great astronomers was not always of an agreeable
character. Kepler s publications are voluminous. His first
attempt to explain the solar system was made in 1596, when he
thought he had discovered a curious relation between the
five regular solids and the number and distance of the planets.
The publication of this pseudo-discovery brought him much
fame. Maturer reflection and intercourse with Tycho Brahe
and Galileo led him to investigations and results more worthy of
his genius "Kepler s laws." He enriched pure mathematics
as well as astronomy. It is not strange that he was interested
in the mathematical science which had done him so much
service; for "if the Greeks had not cultivated conic sections,
Kepler could not have superseded Ptolemy." 11 The Greeks
never dreamed that these curves would ever be of practical
use ; Aristseus and Apollonius studied them merely to satisfy
their intellectual cravings after the ideal; yet the conic
sections assisted Kepler in tracing the march of the planets in
their elliptic orbits. Kepler made also extended use of loga
rithms and decimal fractions, and was enthusiastic in diffusing
a knowledge of them. At one time, while purchasing wine, he
was struck by the inaccuracy of the ordinary modes of deter
mining the contents of kegs. This led him to the study of
the volumes of solids of revolution and to the publication of
the Stereometric Doliorum in 1615. In it he deals first with the >
VIETA TO DESCARTES. 173
areas, volumes, and centres of gravity. He effected the quad
rature of a parabola of any degree y m = a m ~~ l x, and also of a
parabola y m = a m ~ n x n . We have already mentioned his quadra
ture of the cycloid. Eoberval is best known for his method
of drawing tangents. He was the first to apply motion to
the resolution of this important problem. His method is
allied to Newton s principle of fluxions. Archimedes con
ceived his spiral to be generated by a double motion. This
idea Hoberval extended to all curves. Plane curves, as for
instance the conic sections, may be generated by a point
acted upon by two forces, and are the resultant of two
motions. If at any point of the curve the resultant be
resolved into its components, .then the diagonal of the par
allelogram determined by them is the tangent to the curve
at that point. The greatest difficulty connected with this
ingenious method consisted in resolving the resultant into
components having the proper lengths and directions. E,ober-
val did not always succeed in doing this, yet his new idea was
a great step in advance. He broke off from the ancient
definition of a tangent as a straight line having only one point
in common with a curve, a definition not valid for curves of
higher degrees, nor apt even in curves of the second degree to
bring out the properties of tangents and the parts they may
be made to play in the generation of the curves. The subject
of tangents received special attention also from Permat,
Descartes, and Barrow, and reached its highest development
after the invention of the differential calculus. Permat and
Descartes defined tangents as secants whose two points of
intersection with the curve coincide; Barrow considered a
curve a polygon, and called one of its sides produced a tangent.
A profound scholar in all branches of learning and a mathe
matician of exceptional powers was Pierre de Fennat (1601-
1665). He studied law at Toulouse, and in 1631 was made
174 A HISTOEY OF MATHEMATICS.
councillor for the parliament of Toulouse. His leisure time
was mostly devoted to mathematics, which, he studied with
irresistible passion. Unlike Descartes and Pascal, he led a
quiet and unaggressive life. Fermat has left the impress of
his genius upon all branches of mathematics then known. A
great contribution to geometry was his De maximis et minimis.
About twenty years earlier, Kepler had first observed that the
increment of a variable, as, for instance, the ordinate of a
curve, is evanescent for values very near a maximum or a
minimum value of the variable. Developing this idea, Fermat
obtained his rule for maxima and minima. He substituted
x + e for x in the given function of x and then equated to each
other the two consecutive values of the function and divided
the equation by e. If e be taken 0, then the roots of this
equation are the values of x, making the function a maximum
or a minimum. Fermat was in possession of this rule in 1629.
The main difference between it and the rule of the differential
calculus is that it introduces the indefinite quantity e instead
of the infinitely small dx. Fermat made it the basis for his
method of drawing tangents.
Owing to a want of explicitness in statement, Fermat s
method of maxima and minima, and of tangents, was severely
attacked by his great contemporary, Descartes, who could
never be brought to render due justice to his merit. In the
ensuing dispute, Fermat found two zealous defenders in Eober-
val and Pascal, the father; while Midorge, Desargues, and
Hardy supported Descartes.
Since Fermat introduced the conception of infinitely small
differences between consecutive values of a function and ar
rived at the principle for finding the maxima and minima,
it was maintained by Lagrange, Laplace, and Fourier, that
Fermat may be regarded as the first inventor of the differ
ential calculus. This point is not well taken, as will be seen
YIETA TO DESCABTES. 175
from the words of Poisson, himself a Frenchman, who rightly
says that the differential calculus "consists in a system of
rules proper for finding the differentials of all functions,
rather than in the use which may be made of these infinitely
small variations in the solution of one or two isolated prob
lems. 3
A contemporary mathematician, whose genius excelled even
that of the great Fermat, was Blaise Pascal (1623-1662). He
was born at Clermont in Auvergne. In 1626 his father retired
to Paris, where he devoted himself to teaching his son, for he
would not trust his education to others. Blaise Pascal s genius
for geometry showed itself when he was but twelve years
old. His father was well skilled in mathematics, but did not
wish his son to study it until he was perfectly acquainted with
Latin and Greek. All mathematical books were hidden out of
his sight. The boy once asked his father what mathematics
treated of, and was answered, in general, "that it was the
method of making figures with exactness, and of finding out
what proportions they relatively had to one another." He
was at the same time forbidden to talk any more about it,
or ever to think of it. But his genius could not submit to be
confined within these bounds. Starting with the bare fact
that mathematics taught the means of making figures infalli
bly exact, he employed his thoughts about it and with a piece
of charcoal drew figures upon the tiles of the pavement, trying
the methods of drawing, for example, an exact circle or equi
lateral triangle. He gave names of his own to these figures
and then formed axioms, and, in short, came to make perfect
demonstrations. In this way he arrived unaided at the theo
rem that the sum of the three angles of a triangle is equal to
two right angles. His father caught him in the act of study
ing this theorem, and was so astonished at the sublimity and
force of his genius as to weep for joy. The father now gave
176 A HISTORY OF MATHEMATICS.
Mm Euclid s Elements, which, he, without assistance, mastered
easily. His regular studies being languages, the boy employed
only his hours of amusement on the study of geometry, yet he
had so ready and lively a penetration that, at the age of six
teen, he wrote a treatise upon conies, which passed for such a
surprising effort of genius, that it was said nothing equal to it
in strength had been produced since the time of Archimedes.
Descartes refused to believe that it was written by one so
young as Pascal. This treatise "was never published, and is
now lost. Leibniz saw it in Paris and reported on a portion
of its contents. The precocious youth made vast progress in
all the sciences, but the constant application at so tender an
age greatly impaired his health. Yet he continued working,
and at nineteen invented his famous machine for performing
arithmetical operations mechanically. This continued strain
from overwork resulted in a k permanent indisposition, and he
would sometimes say that from the* time he was eighteen,
he never passed a day free from pain. At the age of
twenty-four he resolved to lay aside the study of the human
sciences and to consecrate his talents to religion. His Pro
vincial Letters against the Jesuits are celebrated. But at
times he returned to the favourite study of his youth. Being
kept awake one night by a toothache, some thoughts un-
designedly came into his head concerning the roulette or
cycloid ; one idea followed another ; and he thus discovered
properties of this curve even to demonstration. A corre
spondence between him and Fermat on certain problems was
the beginning of the theory of probability. Pascal s illness
increased, and he died at Paris at the early age of thirty-nine
years. 30 By him the answer to the objection to Cavalieri s
Method of Indivisibles was put in the clearest form. Like
Boberval, he explained " the sum of right lines " to mean. " the
sum of infinitely small rectangles," Pascal greatly advanced
YIETA TO DESCARTES. 177
the knowledge of the cycloid. He determined the area of a
section produced by any line parallel to the base ; the volume
generated by it revolving around its base or around the axis ;
and, finally, the centres of gravity of these volumes, and also
of half these volumes cut by planes of symmetry. Before
publishing his results, he sent, in 1658, to all mathematicians
that famous challenge offering prizes for the first two solu
tions of these problems. Only Wallis and A. La Louere com
peted for them. The latter was quite unequal to the task;
the former, being pressed for time, made numerous mistakes :
neither got a prize. Pascal then published his own solutions,
which produced a great sensation among scientific men. Wal
lis, too, published his, with the errors corrected. Though not
competing for the prizes, Huygens, Wren, and Fermat solved
some of the questions. The chief discoveries of Christopher
Wren (1632-1723), the celebrated architect of Si. Paul s
Cathedral in London, were the rectification of a cycloidal arc
and the determination of its centre of gravity. Fermat
found the area generated by an arc of the cycloid. Huygens
invented the cycloidal pendulum.
The beginning of the seventeenth century witnessed also
a revival of synthetic geometry. One who treated conies still
by ancient methods, but who succeeded in greatly simplifying
many prolix proofs of Apollonius, was Claude Mydorge in
Paris (1585-1647), a friend of Descartes. But it remained
for Girard Desargues (1593-1662) of Lyons, and for Pascal, to
leave the beaten track and cut out fresh paths. They intro
duced the important method of Perspective. All conies on
a cone with circular base appear circular to an eye at the apex.
Hence Desargues and Pascal conceived the treatment of the
conic sections as projections of circles. Two important and
beautiful theorems were given by Desargues : The one is on
the "involution of the six points," in which a transversal
178 A HISTORY OF MATHEMATICS.
meets a conic and an inscribed quadrangle ; the other is that,
if the vertices of two triangles, situated either in space or in
a plane, lie on three lines meeting in a point, then their sides
meet in three points lying, on > Hnej and conversely. This
last theorem has been employed in recent times by Branchion,
Sturm, Gergonne, and Poncelet. Poncelet made it the basis
of his beautiful theory of hoinoligical figures. We owe to
Desargues the theory of involution and of transversals ; also
the beautiful conception that the two extremities of a straight
line may be considered as meeting at infinity, and that paral
lels differ from other pairs of lines only in having their points
of intersection at infinity. Pascal greatly admired Desargues
results, saying (in his Essais pour les Coniques), "I wish to
acknowledge that I owe the little that I have discovered on
this subject, to his writings." PascaTsTnd Desargues 3 writ
ings contained the fundamental ideas of modern synthetic
geometry. In Pascal s wonderful work on conies, written
at the age of sixteen and now lost, were given the theorem
on the anharmonic ratio, first found in Pappus, and also that
celebrated proposition on the mystic hexagon, known as
"Pascal s theorem," viz. that the opposite sides of a hexagon
inscribed in a conic intersect in three points which are col-
linear. This theorem formed the keystone to his theory. He
himself said that from this alone he deduced over 400 corol
laries, embracing the conies of Apollonius &nd many other
results. Thus the genius of Desargues and Pascal uncovered
several of the rich treasures of modern synthetic geometry;
but owing to the absorbing interest taken in the analytical
geometry of Descartes and later in the differential calculus,
the subject was almost entirely neglected until the present
century.
In the theory of numbers no new results of scientific value
had been reached for over 1000 years, extending from the
DESCARTES TO NEWTOH. 183
heavier ; he established the first law of motion ; determined
the laws of falling bodies ; and, having obtained a clear notion
of acceleration and of the independence of different motions,
was able to prove that projectiles move in parabolic curves.
Up to his time it was believed that a cannon-ball moved
forward at first in a straight line and then suddenly "fell
vertically to the ground. Galileo had an understanding of
centrifugal forces, and gave a correct definition of momentum.
Though he formulated the fundamental principle of statics,
known as the parallelogram of forces, yet he did not fully
recognise its scope. The principle of virtual velocities was
partly conceived by Guido Ubaldo (died 1607), and afterwards
more fully by Galileo.
Galileo is the founder of the science of dynamics. Among
his contemporaries it was chiefly the novelties he detected in
the sky that made him celebrated, but Lagrange claims that
his astronomical discoveries required only a telescope and
perseverance, while it took an extraordinary genius to dis
cover laws from phenomena, which we see constantly and of
which the true explanation escaped all earlier philosophers.
The first contributor to the science of mechanics after Galileo
was Descartes.
DESCABTES TO NEWTOK
Among the earliest thinkers of the seventeenth and eigh
teenth centuries, who employed their mental powers toward the
destruction of old ideas and the up-building of new ones, ranks
Rene Descartes (1596-1650). Though he professed orthodoxy
in faith all his life, yet in science he was a profound sceptic.
He found that the world s brightest thinkers had been long
exercised in metaphysics, yet they had discovered nothing
184 A HISTORY OF MATHEMATICS.
certain; nay, had even flatly contradicted each other. This
led him to the gigantic resolution of taking nothing whatever
on authority, but of subjecting everything to scrutinous exam
ination, according to new methods of inquiry. The certainty
of the conclusions in geometry and arithmetic brought out in
his mind the contrast between the true and false ways of
seeking the truth. He thereupon attempted to apply mathe-*
matical reasoning to all sciences. " Comparing the mysteries
of nature with the laws of mathematics, he dared to hope that
the secrets of both could, be unlocked with the same key."
Thus he "built up a system of philosophy called Cartesianism.
Great as was Descartes celebrity as a metaphysician, it
may be fairly questioned whether his claim to be remembered
by posterity as a mathematician is not greater. His philosophy
has long since been superseded by other systems, but the ana
lytical geometry of Descartes will remain a valuable possession
forever. At the age of twenty-one, Descartes enlisted in the
army of Prince Maurice of Orange. His years of soldiering
were years of leisure, in which he had time to pursue his
studies. At that time mathematics was his favourite science.
But in 1625 he ceased to devote himself to pure mathematics.
Sir William. Hamilton is in error when he states that Descartes
considered mathematical studies absolutely pernicious as a
means of internal culture. In a letter to Mersenne, Descartes
says : " M. Desargues puts me under obligations on account of
the pains that it has pleased him to have in me, in that he
shows that he is sorry that I do not wish to study more in geom
etry, but I have resolved to quit only abstract geometry, that
is to say, the consideration of questions which serve only to
exercise the mind, and this, in order to study another kind of
geometry, which has for its .object the explanation of the
phenomena of nature. . . . You know that all my physics is
nothing else than geometry. 77 The years between 1629 and
DESCARTES TO NEWTON". 185
1649 were passed by him in Holland in the study, principally,
of physics and metaphysics. His residence in Holland was
during the most brilliant days of the Dutch state. In 1637 he
published his Discours de la Methode> containing among others
an essay of 106 pages on geometry. His Geometry is not easy
reading. An edition appeared subsequently with notes by his
friend De Beaune, which were intended to remove the dif
ficulties.
It is frequently stated that Descartes was the first to apply
algebra to geometry. This statement is inaccurate, for Yieta
and others had done this before him. Even the Arabs some
times used algebra in connection with geometry. The new
step that Descartes did take was the introduction into geom
etry of an analytical method based on the notion of variables
and constants, which enabled him to represent curves by alge
braic equations. In the Greek geometry, the idea of motion
was wanting, but with Descartes it became a very fruitful
conception. By him a point on a plane was determined in
position by its distances from two fixed right lines or axes.
These distances varied with every change of position in
the point. This geometric idea of co-ordinate representation,
together with the algebraic idea of two variables in one equa
tion having an indefinite number of simultaneous values, fur
nished a method for the study of loci, which is admirable for
the generality of its solutions. Thus the entire conic sections
of Apollonius is wrapped up and contained in a single equa
tion of the second degree.
The Latin term for "ordite>" used by Descartes comes
from the expression linece ordinatce, employed by Eoman sur
veyors for parallel lines. The term abscissa occurs for the
first time in a Latin work of 1659, written by Stefano degli
Angeli (1623-1697), a professor of mathematics in Rome. 3
Descartes geometry was called "analytical geometry/ 3 partly
186 A HISTORY OF MATHEMATICS.
because, unlike the synthetic geometry of tie ancients, it is
actually analytical, in the sense that the word is used in
logic ; and partly because the practice had then already
arisen, of designating by the term analysis the calculus with
general quantities.
The first important example solved by Descartes in his
geometry is the " problem of Pappus " ; viz. " Given several
straight lines in a plane, to find the locus of a point such that
the perpendiculars, or more generally, straight lines at given
angles, drawn from the point to the given lines, shall satisfy
the condition that the product of certain of them shall be in
a given ratio to the product of the rest." Of this celebrated
problem, the Greeks solved only the special case when the
number of given lines is four, in which case the locus of the
point turns out to be a conic section. By Descartes it was
solved completely, and it afforded an excellent example of
the use which can be made of his analytical method in the
study of loci. Another solution was given later by Newton
in the Principia.
The methods of drawing tangents invented by Boberval
and Fermat were noticed earlier. Descartes gave a third
method. Of all the problems which he solved by his geometry,
none gave him as great pleasure as his mode of constructing
tangents. It is profound but operose, and, on that account,
inferior to Fermat s. His solution rests on the method of
Indeterminate Coefficients, of which he bears the honour of
invention. Indeterminate coefficients were employed by him
also in solving bi-quadratic equations.
The essays of Descartes on dioptrics and geometry were
sharply criticised by Fermat, who wrote objections to the
former, and sent his own treatise on " maxima and minima "
to show that there were omissions in the geometry. Descartes
thereupon made an attack on Fermat s method of tangents.
DESCARTES TO NEWTON. 187
Descartes was in the wrong in tins attack, yet he continued
the controversy with obstinacy. He had a controversy also
with B-oberval on the cycloid. This curve has been called
the " Helen of geometers," on account of its beautiful proper
ties and the controversies which their discovery occasioned.
Its quadrature by Eoberval was generally considered a brill
iant achievement, but Descartes commented on it by saying
that any one moderately well versed in geometry might have
done this. He then sent a short demonstration of his own.
On Boberval s intimating that he had been assisted by a
knowledge of the solution, Descartes constructed the tangent
to the curve, and challenged Roberval and Fermat to do the
same. ITermat accomplished it, but Eoberval never succeeded
in solving this problem, which had cost the genius of Des
cartes but a moderate degree of attention.
He studied some new curves, now called " ovals of. Des
cartes," which were intended by him to serve in the con
struction of converging lenses, but which yielded no results
of practical value.
The application of algebra to the doctrine of curved lines
reacted favourably upon algebra. As an abstract science,
Descartes improved it by the systematic use of exponents and
by the full interpretation and construction of negative quanti
ties. Descartes also established some theorems on the theory of
equations. Celebrated is his " rule of signs " for determining
the number of positive and negative roots ; viz. an equation
may have as many + roots as there are variations of signs, and
as many roots as there are permanencies of signs. Descartes
was charged by Wallis with availing himself, without acknowl
edgment, of Harriot s theory of equations, particularly his mode
of generating equations ; but there seems to be no good ground
for the charge. Wallis also claimed that Descartes failed to
observe that the above rule of signs is not true whenever the
188 A HISTORY OF MATHEMATICS.
equation has imaginary roots ; but Descartes does not say that
the equation always has, but that it may have so many roots.
It is true that Descartes does not consider the case of irnagi-
naries directly, but further on in * his Geometry he gives
incontestable evidence of being able to handle this case
also.
In mechanics, Descartes can hardly be said to have advanced
beyond Galileo. The latter had overthrown the ideas of
Aristotle on this subject, and Descartes simply " threw himself
upon the enemy " that had already been " put to the rout."
His statement of the first and second laws of motion was an
improvement in form ; but his third law is false in substance.
The motions of bodies in their direct impact was imperfectly
understood by Galileo, erroneously given by Descartes, and
first correctly stated by Wren, "Wallis, and Huygens.
One of the most devoted pupils of Descartes was the learned
Princess Elizabeth, daughter of Frederick Y. She applied the
new analytical geometry to the solution of the " Apollonian
problem." His second royal follower was Queen Christina,
the daughter of Gustavus Adolphus. She urged upon Des
cartes to come to the Swedish court. After much hesitation
he accepted the invitation in 1649. He died at Stockholm one
year later. His life had been one long warfare against the
prejudices of men.
It is most remarkable that the mathematics and philosophy
of Descartes should at first have been appreciated less by his
countrymen, than by foreigners. The indiscreet temper of
Descartes alienated the great contemporary French mathema
ticians, Roberval, Ferrnat, Pascal. They continued in investi
gations of their own, and on some points strongly opposed
Descartes. The universities of France were under strict
ecclesiastical control and did nothing to introduce his mathe
matics and philosophy. It was in the youthful universities of
DESCARTES TO NEWTON. 189
Holland that the effect of Cartesian teachings was most
immediate and strongest.
The only prominent Frenchman who immediately followed
in the footsteps of the great master was De Beaune (1601-1652).
He was one of the first to point out that the properties of a
curve can be deduced from the properties of its tangent. This
mode of inquiry has been called the inverse method of tangents.
He contributed to the theory of equations by considering for
the first time the upper and lower limits of the roots of
numerical equations.
In the Netherlands a large number of distinguished mathema
ticians were at once struck with admiration for the Cartesian
geometry. Foremost among these are van Schooten, John de
Witt y van Heuraet, Sluze, and Hudde. Van Schooten (died 1660),
professor of mathematics at Leyden, brought out an edition
of Descartes geometry, together with the notes thereon by
De Beaune. His chief work is his Hxercitationes Mathematics,
in which he applies the analytical geometry to the solution of
many interesting and difficult problems. The noble-hearted
Johann de "Witt, grand-pensioner of Holland, celebrated as a
statesman and for his tragical end, was an- ardent geometrician.
He conceived a new and ingenious way of generating conies,
which is essentially the same as -that by protective pencils of
rays in modern synthetic geometry. He treated the subject
not synthetically, but with aid of the Cartesian analysis.
Rene Francois de Sluze (1622-1685) and Johann Hudde (1633-
1704) made some improvements on Descartes and Fermat s
methods of drawing tangents, and on the theory of maxima and
minima. With Hudde, we find the first use of three variables
in analytical geometry. He is the author of an ingenious rule
for finding equal roots. We illustrate it by the equation,
cc 3 . as 2 8 # -}- 12 = 0.^ Taking an arithmetical progression
3, 2, 1, 0, of which the highest term is equal to the degree of
190 A HISTOEY OF MATHEMATICS.
the equation, and multiplying each term of the equation respec
tively by the corresponding term of the progression, we get
3^-2^-8^ = 0, or 3a 2 2x 8 = 0. This last equation
is by one degree lower than the original one. [Find the G.C.D.
of the two equations. This is x 2 ; hence 2 is one of the two
equal roots. Had there been no common divisor, then the
original equation would not have possessed equal roots. Hudde
gave a demonstration for this rule. 24
Heinrich van Heuraet must be mentioned as one of the earli
est geometers who occupied themselves with success in the.
rectification of carves. He observed in a general way that the
two problems of quadrature and of rectification are really
identical, and that the one can be reduced to the other. Thus
he carried the rectification of the hyperbola back to the
quadrature of the hyperbola. The semi-cubical parabola
y 8 = ao? 2 was the first curve that was ever rectified absolutely.
This appears to have been accomplished independently by Van
Heuraet in Holland and by William Neil (1637-1670) in Eng
land. According to Wallis the priority belongs to ISfeil. Soon
after, the cycloid was rectified by "Wren and Fermat.
The prince of philosophers in Holland, and one of the
greatest scientists of the seventeenth century, was Christian
Huygens (1629-1695), a native of the Hague. Eminent as a
physicist and astronomer, as well as mathematician, he was
a worthy predecessor of Sir Isaac Newton. He studied at
Ley den under the younger Van ScJwoten. The perusal of
some of his earliest theorems led Descartes to predict his
future greatness. In 1651 Huygens wrote a treatise in which
he pointed out the fallacies of Gregory St. Vincent (1584-1667)
on the subject of quadratures. He himself gave a remarkably
close and convenient approximation to the length of a circular
arc. In 1660 and 1663 he went to Paris and to London. In
1666 he was appointed by Louis XIV. member of the French
DESCARTES TO NEWTON. 191
Academy of Sciences. He "was induced to remain in Paris
from that time until 1681, when he returned to his native
city, partly for consideration of his health and partly on
account of the revocation of the Edict of Nantes.
The majority of his profound discoveries were made with
aid of the ancient geometry, though at times he used the
geometry of Descartes or of Cavalieri and Fermat. Thus,
like his illustrious friend, Sir Isaac Newton, he always showed
partiality for the Greek geometry. Newton and Huygens
were kindred minds, and had the greatest admiration for each
other. Newton always speaks of him as the "Summus
Hugenius."
To the two curves (cubical parabola and cycloid) previously
rectified he added a third, the cissoid. He solved the
problem of the catenary, determined the surface of the
parabolic and hyperbolic conoid, and discovered the proper
ties of the logarithmic curve and the solids generated by it.
Huygens De Jwrologio osdllatorio (Paris, 1673) is a work that
ranks second only to the Principia of Newton and constitutes
historically a necessary introduction to it. 13 The book opens
with a description of pendulum clocks, of which Huygens
is the inventor. Then follows a treatment of accelerated
motion of bodies falling free, or sliding on inclined planes, or
on given curves, culminating in the brilliant discovery that
the cycloid is the tautochronous curve. To the theory of
curves he added the important theory of "evolutes." After
explaining that the tangent of the eyolute is normal to the
involute, he applied the theory to the cycloid, and showed by
simple reasoning that the evolute of this curve is an equal
cycloid. Then comes the complete general discussion of the
centre of oscillation. This subject had been proposed for
investigation by Mersenne and discussed by Descartes and
Eoberval. In Huygens assumption that the common centre
192 A HISTORY OF MATHEMATICS.
of gravity of a group of bodies, oscillating about a horizontal
axis, rises to its original height, but no higher, is expressed
for the first time one of the most beautiful principles of
dynamics, afterwards called the principle of the conservation
of vis viva* 2 The thirteen theorems at the close of the work
relate to the theory of centrifugal force in circular motion.
This theory aided Newton in discovering the law of gravita
tion.
Huygens wrote the first formal treatise on probability. He
proposed the wave-theory of light and with great skill applied
geometry to its development. This theory was long neglected,
but was revived and successfully worked out by Young and
Fresnel a century later. Huygens and his brother improved
the telescope by devising a better way of grinding and polish
ing lenses. With more efficient instruments he determined
the nature of Saturn s appendage and solved other astro
nomical questions. Huygens Opuscula posthuma appeared
in 1703.
Passing now from Holland to England, we meet there one
of the most original mathematicians of his day John Wallis
(1616-1703). He was educated for the Church at Cambridge
and entered Holy Orders. But his genius was employed
chiefly in the study of mathematics. In 1649 he was appointed
Savilian professor of geometry at Oxford. He was one of
the original members of the Eoyal Society, which was founded
in 1663. Wallis thoroughly grasped the mathematical methods
both of Cavalieri and Descartes. His Conic Sections is the
earliest work in which these curves are no longer considered
as sections of a cone, but as curves of the second degree, and
are treated analytically by the Cartesian method of co-or
dinates. In this work Wallis speaks of Descartes in the
highest terms, but in his Algebra he, .without good reason,
accuses Descartes of plagiarising from Harriot. We have
KEWTOK TO ETJLER. 199
But a : e = the ordinate : the sub-tangent ; hence
p : 2 Vpx = Vp# : sub-tangent, /
giving 2 x for the value of the sub-tangent.^this method dif
fers from that of the differential calculus only in notation. 31
NEWTON TO EULEE.
It has been seen that in France prodigious scientific progress
was made during the beginning and middle of the seventeenth
century. The toleration which marked the reign of Henry IV.
and Louis XIII. was accompanied by intense intellectual
activity. Extraordinary confidence came to be placed in the
power of the human mind. The bold intellectual conquests
of Descartes, Fermat, and Pascal enriched mathematics with
imperishable treasures. During the early part of the reign
of Louis XIV. we behold the sunset splendour of this glorious
period. Then followed a night of mental effeminacy. This
lack of great scientific thinkers during the reign of Louis XIV.
may be due to the simple fact that no great minds were born ;
but, according to Buckle, it was due to the paternalism, to
the spirit of dependence and subordination, and to the lack
of toleration, which marked the policy of Louis XIV.
In the absence of great French thinkers, Louis XIV. sur
rounded himself by eminent foreigners. Bonier from Den
mark, Huygens from Holland, Dominic Cassini from Italy,
were the mathematicians and astronomers adorning Ms court.
They were in possession of a brilliant reputation before going
to Paris. Simply because they performed scientific work in
Paris, that work belongs no more to France than the dis
coveries of Descartes belong to Holland, or those of Lagrange
to Germany, or those of Euler and Poncelet to Eussia. We
200 A HISTORY OF MATHEMATICS.
must look to other countries than "France for the great scien
tific men of the latter part of the seventeenth century.
About the time when Louis XIV. assumed the direction
of the French government Charles II. became king of Eng
land. At this time England was extending her commerce
and navigation, and advancing considerably in material pros
perity. A strong intellectual movement took place, which
was unwittingly supported by the king. The age of poetry
was soon followed by an age of science and philosophy. In
two successive centuries England produced Shakespeare and
Newton !
Germany still continued in a state of national degradation.
The Thirty Years 5 War had dismembered the empire and
brutalised the people. Yet this darkest period of Germany s
history produced Leibniz, one of the greatest geniuses of
modern times.
There are certain focal points in history toward which
the lines of past progress converge, and from which radiate
the advances of the future. Such was the age of Newton
and Leibniz in the history of mathematics. During fifty
years preceding this era several of the brightest and acutest
mathematicians bent the force of their genius in a direction
which finally led to the discovery of the infinitesimal calculus
by Newton and Leibniz. Cavalieri, Roberval, Fermat, Des
cartes, Wallis, and others had each contributed to the new
geometry. So great was the advance made, and so near
was their approach toward the invention of the infinitesimal
analysis, that both Lagrange and Laplace pronounced their
countryman, Fermat, to be the true inventor of it. The dif
ferential calculus, therefore, was not so much an individual
discovery as the grand result of a succession of discoveries
by different minds. Indeed, no great discovery ever flashed
upon the mind at once, and though those of Newton will
HEWTON TO EULBR. 201
influence mankind to the end of the world, yet it must be
admitted that Pope s lines are only a " poetic fancy " :
" Nature and Nature s laws lay hid in night ;
God said, c Let Newton be, and all was light."
Isaac Newton (1642-1727) was born at Woolsthorpe, in
Lincolnshire; the same year in which Galileo died. At his
birth he was so small and weak that his life was despaired of.
His mother sent him at an early age to a village school, and
in his twelfth year to the public school at G-rantham. At
first he seems to have been very inattentive to his studies
and very low in the school; but when, one day, the little
Isaac received a severe kick "upon Ms stomach from a boy
who was above him, he laboured hard till he ranked higher
in school than his antagonist. From that time he continued
to rise until he was the head boy. 33 At Grantham, Isaac
showed a decided taste for mechanical inventions. He con
structed a water-clock, a wind-mill, a carriage moved by the
person who sat in it, and other toys. When he had attained
his fifteenth year his mother took him home to assist her in
the management of the farm, but his great dislike for farm-
work and his irresistible passion for study, induced her to
send him back to Grantham, where he remained till his
eighteenth year, when he entered Trinity College, Cambridge
(1660). Cambridge was the real birthplace of Newton s
genius. Some idea of his strong intuitive powers may be
drawn from the fact that he regarded the theorems of ancient
geometry as self-evident truths, and that, without any prelimi
nary study, he made himself master of Descartes Geometry.
He afterwards regarded this neglect of elementary geometry
a mistake in his mathematical studies, and he expressed to
Dr. Pemberton his regret that "he had applied himself to the
works of Descartes and other algebraic writers before he had
202 A HISTOBY O^ MATHEMATICS.
considered the Elements of Euclid with, that attention which
so excellent a writer deserves." Besides Descartes Geometry,
he studied Oughtred s Clams, Kepler s Optics, the works of
Vieta, Schooten s Miscellanies, Barrow s Lectures, and the
works of Wallis. He was particularly delighted with Wallis
Arithmetic of Infinites, a treatise fraught with rich and varied
suggestions. Newton had the good fortune of having for
a teacher and fast friend the celebrated Dr. Barrow, who
had been elected professor of Greek in 1660, and was made
Lucasian professor of mathematics in 1663. The mathe
matics of Barrow and of Wallis were the starting-points
from which Newton, with a higher power than his masters 3 ,
moved onward into wider fields. Wallis had effected the
quadrature of curves whose ordinates are expressed by any
integral and positive power of (1 # 2 ). We have seen how
Wallis attempted but failed to interpolate between the areas
thus calculated, the areas of other curves, such as that of
the circle; how Newton attacked the problem, effected the
interpolation, and discovered the Binomial Theorem, which
afforded a much easier and direct access to the quadrature
of curves than did the method of interpolation; for even
though the binomial expression for the ordinate be raised
to a fractional or negative power, the binomial could at once
be expanded into a series, and the quadrature of each separate
term of that series could be effected by the method, of Wallis.
Newton introduced the system of literal indices." "
Newton s study of quadratures soon led him to another
and most profound invention. He himself says that in 1665
and 1666 he conceived the method of fluxions and applied
them to the quadrature of curves. Newton did not com
municate the invention to any of his friends till 1669, when
lie placed in the hands of Barrow a tract, entitled De Analyst
per ^Equationes Numero Terminorum Infinitas, which was sent
NEWTON TO ETJLER. 203
by Barrow to Collins, who greatly admired it. In this treatise
the principle of fluxions, though distinctly pointed out, is only
partially developed and explained. Supposing the abscissa
to increase uniformly in proportion to the time, he looked
upon the area of a curve as a nascent quantity increasing
by continued fluxion in the proportion of the length of the
ordinate. The expression which was obtained for the fluxion
he expanded into a finite or infinite series of monomial terms,
to which Wallis rule was applicable. Barrow urged Newton
to publish this treatise 5 " but the modesty of the author, of
which the excess, if not cxilpable, was certainly in the present
instance very unfortunate, prevented his compliance." 26 Had
this tract been published then, instead of forty-two years
later, there would probably have been no occasion for that
long and deplorable controversy between Newton and Leibniz.
ITor a long time Newton s method remained unknown, ex
cept to his friends and their correspondents. In a letter
to Collins, dated December 10th, 1672, Newton states the fact
of his invention with one example, and then says : " This
is one particular, or rather corollary, of a general method,
which extends itself, without any troublesome calculation, not
only to the drawing of tangents to any curve lines, whether
geometrical or mechanical, or anyhow respecting right lines r
or other curves, but also to the resolving other abstruser
kinds of problems about the crookedness, areas, lengths,
centres of gravity of curves, etc.; nor is it (as Hudden s
method of Maximis and Minimis) limited to equations which
are free from surd quantities. This method I ha ve inter
woven with that other of working in equations, by reducing
them to infinite series."
These last words relate to a treatise he composed in the
year 1671, entitled Method of Fluxions, in which he aimed
to represent Ms method as an independent calculus and as
204 A HISTORY OF MATHEMATICS.
a complete system. This tract was intended as an introduc
tion to an edition of Kinckhuysen s Algebra, which he had
undertaken to publish. " But the fear of being involved in
disputes about this new discovery, or perhaps the wish to
render it more complete, or to have the sole advantage of
employing it in his physical researches, induced him to aban
don this design."
Excepting two papers on optics, all of his works appear
to have been published only after the most pressing solicita
tions of his friends and against his own wishes. 34 His re
searches on light were severely criticised, and he wrote in
1675: "I was so persecuted with discussions arising out of
my theory of light that I blamed my own imprudence for
parting with so substantial a blessing as my quiet to run
after a shadow."
The Method of Fluxions, translated by J. Colson from New
ton s Latin, was first published in 1736, or sixty-five years
after it was written. In it he explains, first the expansion
into series of fractional and irrational quantities, a stibject
which, in his first years of study, received the most careful
attention. He then proceeds to the solution of the two fol
lowing mechanical problems, which constitute the pillars, so
to speak, of the abstract calculus :
"I. The length of the space described being continually
(i.e. at all times) given ; to find the velocity of the motion at
any time proposed,
" II. The velocity of the motion being continually given ; to
find the length of the space described at any time proposed."
Preparatory to the solution, Newton says : " Thus, in the
equation y = x 2 , if y represents the length of the space at any
time described, which (time) another space x, by increasing
with an uniform celerity x, measures and exhibits as described:
then 2 xx will represent the celerity by which the space y,
NEWTON TO EtJLEB. 205
at the same moment of time, proceeds to be described ; and
contrarywise."
" But whereas we need not consider the time here, any far
ther than it is expounded and measured by an equable local
motion; and besides, whereas only quantities of the same
kind can be compared together, and also their velocities of
increase and decrease ; therefore, in what follows I shall have
no regard to time formally considered, but I shall suppose
some one of the quantities proposed, being of the same kind,
to be increased by an equable fluxion, to which the rest may be
referred, as it were to time ; and, therefore, by way of analogy,
it may not improperly receive the name of time." In this
statement of Newton there is contained a satisfactory answer
to the objection which has been raised against his method,
that it introduces into analysis the foreign idea of motion. A
quantity thus increasing by uniform fluxion, is what we now
call an independent variable.
Newton continues : " Now those quantities which I consider
as gradually and indefinitely increasing, I shall hereafter call
fluents, or flowing quantities, and shall represent them by the
final letters of the alphabet, v, x, y, and z ; . . . and the veloci
ties by which every fluent is increased by its generating motion
(which I may call fluxions, or simply velocities, or celerities),
I shall represent by the same letters pointed, thus, v, x, y, z.
That is, for the celerity of the quantity v I shall put v, and so
for the celerities of the other quantities x, y, and z, I shall put
x, y, and z, respectively. 57 It must here be observed that New
ton does not take the fluxions themselves infinitely small.
The " moments of fluxions," a term introduced further on, are
infinitely small quantities. These " moments," as defined and
used in the Method of Fluxions, are substantially the differen
tials of Leibniz. De Morgan points out that no small amount of
confusion has arisen from the use of the word fluxion and the
206 A HISTORY OF MATHEMATICS.
notation x by all the English writers previous to 1704, except
ing Newton and Cheyne, in the sense of an infinitely small in
crement. 35 Strange to say, even in the Gommercium Eplstolicum
the words moment and fluent appear to be used as synonymous.
After showing by examples how to solve the first problem ;
Newton proceeds to the demonstration of his solution :
"The moments of flowing quantities (that is, their indefi
nitely small parts, by the accession of which, in infinitely
small portions of time, they are continually increased) are as
the velocities of their flowing or increasing.
" Wherefore, if the moment of any one (as x) be represented
by the product of its celerity x into an infinitely small quantity
(i.e. by xty, the moments of the others, V, y, z, will be repre
sented by 0, $0, zO-j because i)0, xQ, 0, and zO are to each other
as v, x, y, and z.
" Now since the moments, as xQ and $0, are the indefinitely
little accessions of the flowing quantities x and y, by which
those quantities are increased through the several indefinitely
little intervals of time, it follows that those quantities, x
and y, after any indefinitely small interval of time, become
x -4- xO and y + 2/0, and therefore the equation, which at all
times indifferently expresses the relation of the flowing quan
tities, will as well express the relation between x + xO and
y + $0, as between x and y ; so that x + xO and y -j- $0 may
be substituted in the same equation for those quantities, in
stead of x and y. Thus let any equation X s ax 2 + axy y B =
be given, and substitute x + xO for x, and y + yQ for y, and
there will arise
a? 3 + 3a; 2 a;0 + SxxQxQ + 3 3
ax* 2 axxO axOxO
+ axy + ayxQ + a0#0 = 0.
NEWTON TO EtJLEJB. 207
" Now, by supposition, $ ax? + axy y 5 = 0, which there
fore, being expunged and the remaining terms being divided
by 0, there will remain
3x 2 x 2 axx + ayx -f axy 3y*y + 3 #&cO axdto + a�
3 yyyQ + 3()0 ^00 = 0.
But whereas zero is supposed to be infinitely little, that it
may represent the moments of quantities, the terms that are
.multiplied by it will be nothing in respect of the rest (termini
in earn ducti pro niliilo possunt liaberi cum aliis collati) ; there
fore I reject them, and there remains
3x 2 x 2 axx + ayx + axy 3y 2 y = 0,
as above in Example I." Newton here uses infinitesimals.
Much greater than in the first problem were the difficulties
encountered in the solution of the second problem, involving,
as it does, inverse operations which have been taxing the skill
of the best analysts since his time. Newton gives first a
special solution to the second problem in which he resorts
to a rule for which he has given no proof.
In the general solution of his second problem, Newton
assumed homogeneity with respect to the fluxions and then
considered three cases : (1) when the equation contains two
fluxions of quantities and but one of the fluents; (2) when
the equation involves both the fluents as well as both the flux
ions ; (3) when the equation contains the fluents and the flux
ions of three or more quantities. The first case is the easiest
since it requires simply the integration of -^=/(a;), to which
ax
his "special solution" is applicable. The second case de
manded nothing less than the general solution of a dif
ferential equation of the first order. Those who know what
efforts were afterwards needed for the complete exploration
of this field in analysis, will not depreciate Newton s work
208 A HISTOEY OF MATHEMATICS.
even though, he resorted to solutions in form of infinite series.
Newton s third case comes now under the solution of partial
differential equations. He took the equation 2 a* z -f xy =
and succeeded in finding a particular integral of it.
The rest of the treatise is devoted to the determination of
maxima and minima, the radius of curvature of curves, and
other geometrical applications of his fluxionary calculus. All
this was done previous to the year 1672.
It must be observed that in the Method of Fluxions (as well
as in his De Analysi and all earlier papers) the method
employed by Newton is strictly infinitesimal, and in substance
like that of Leibniz. Thus, the original conception of the
calculus in England, as well as on the Continent, was based
on infinitesimals. The fundamental principles of the fluxionary
calculus were first given to the world in the Principia; but its
peculiar notation did not appear until published in the second
volume of "Wallis Algebra in 1693. The exposition given in
the Algebra was substantially a contribution of Newton; it
rests on infinitesimals. In the first edition of the Principia
(1687) the description of fluxions is likewise "founded on
infinitesimals, but in the second (1713) the foundation is
somewhat altered. In Book II. Lemma II. of the first edition
we read : " Cave tamen intellexeris particulas finitas. Momenta
quam primum finitce sunt magnitudiniSj desinunt esse momenta.
Finiri enim repugned aliquatenus perpetuo eorum incremento vel
decremento. Intelligenda sunt principia jamjam nascentia
finitorum magnitudinum." In the second edition the two
sentences which we print in italics are replaced by the
following: "Particulae finitse non sunt momenta sed quanti-
tates ipsse ex momentis genitse." Through the difficulty of
the phrases in both extracts, this much distinctly appears,
that in the first, moments are infinitely small quantities.
What else they are in the second is not clear. 85 In the
NEWTON TO EULEB. 209
Quadrature of Curves of 1704, the infinitely small quantity is
completely abandoned. It lias been shown that in the Method
of Fluxions Newton rejected terms involving the quantity 0,
because they are infinitely small compared with other terms.
This reasoning is evidently erroneous ; for as long as is a
quantity, though ever so small, this rejection cannot be made
without affecting the result. Newton seems to have felt this,
for in the Quadrature of Curves he remarked that " in math
ematics the minutest errors are not to be neglected" (errores
quam minimi in rebus mathematicis non sunt contemnendi) .
The early distinction between the system of Newton and
Leibniz lies in this, that Newton, holding to the conception
of velocity or fluxion, used the infinitely small increment as a
means of determining it, while with Leibniz the relation of the
infinitely small increments is itself the object of determination.
The difference between the two rests mainly upon a difference
in the mode of generating quantities. 35
We give Newton s statement of the method of fluxions or
rates, as given in the introduction to his Quadrature of Curves.
"I consider mathematical quantities in this place not as
consisting of very small parts, but as described by a continued
motion. Lines are described, and thereby generated, not by
the apposition of parts, but by the continued motion of points ;
superficies by the motion of lines; solids by the motion of
superficies ; angles by the rotation of the sides ; portions of
time by continual flux : and so on in other quantities. These
geneses really take place in the nature of things, and are
daily seen in the motion of bodies. . . .
"Eluxions are, as near as we please (quam proxime), as the
increments of fluents generated in times, equal and as small as
possible, and to speak accurately, they are in the prime ratio
of nascent increments ; yet they can be expressed by any lines
whatever, which are proportional to them."
210
A HISTORY OF MATHEMATICS.
T X* 3
Newton exemplifies this last assertion by the problem of
tangency : Let AB be the abscissa, BO the ordinate, VCH
the tangent, EC the increment of the ordinate, which pro
duced meets FjETat T, and Oc the increment of the curve.
The right line Oc being produced to K, there are formed
three small triangles, the rectilinear GEc, the mixtilinear
CEc, and the rectilinear GET. Of these, the first is evidently
the smallest, and the last the greatest. Now suppose
the ordinate be to move- into the place BO, so that the
point c exactly co
incides with the
point (7; OK, and
therefore the curve
Cc, is coincident
with the tangent
OH, EC is abso
lutely equal to E T,
and the mixtilinear
evanescent triangle CEc is, in the last form, similar to the
triangle GET, and its evanescent sides GE, EC, Cc } will be
proportional to CE, ET, and CT, the sides of the triangle
GET. Hence it follows that the fluxions of the lines AB,
BG, AO, being in the last ratio of their evanescent increments,
are proportional to the sides of the triangle GET, or, which ia
all one, of the triangle VBO similar thereunto. As long as
the points G and c are distant from each other by an interval,
however small, the line OJt" will stand apart by a small angle
from the tangent CH. But when CK coincides with GH, and
the lines GE, EC, cG reach their ultimate ratios, then the
points G and c accurately coincide and are one and the same.
Newton then adds that et in mathematics the minutest errors
are not to be neglected." This is plainly a rejection of the
postulates of Leibniz. The doctrine of infinitely small quan-
NEWTON TO EULER. 211
titles is here renounced in a manner which, would lead one
to suppose that Newton had never held it himself. Thus it
appears that Newton s doctrine was different in different
periods. Though, in the above reasoning, the Charybdis of
infinitesimals is safely avoided, the dangers of a Scylla stare
us in the face. We are required to believe that a point may
be considered a triangle, or that a triangle can be inscribed
in a point ; nay, that three dissimilar triangles become similar
and equal when they have reached their ultimate form in one
and the same point.
In the introduction to the Quadrature of Curves the fluxion
of x n is determined as follows :
" In the same time that x, by flowing, becomes x + 0, the
powers 71 becomes (#+0) n , i.e. by the method of infinite series
x n + nQ x"" 1 + ^-^ O 2 x n ~* + etc,,
and the increments
o
and nQ x n ~ l + n ~ ~~ n O 2 x n ~ 2 + etc.,
2
are to one another as
1 to nx"- 1 + ^=2 x n ~ 2 + etc.
"Let now the increments vanish, and their last proportion
will be 1 to nx n ~ l : hence the fluxion of the quantity x is to
the fluxion of the quantity x n as 1 : nx n ~~ l .
" The fluxion of lines, straight or curved, in all cases what
ever, as also the fluxions of superficies, angles, and other
quantities, can be obtained in the same manner by the method
of prime and ultimate ratios. But to establish in this way
the analysis of infinite quantities, and to investigate prime
and ultimate ratios of finite quantities, nascent or evanescent,
is in harmony with the geometry of the ancients ; and I have
endeavoured to show that, in the method of fluxions, it is not
212, A HISTORY OF MATHEMATICS.
necessary to introduce into geometry infinitely small quanti
ties." This mode of differentiating does not remove all the
difficulties connected with, the subject. When becomes
nothing, then we get the ratio - = nx n ~ l , which needs further
elucidation. Indeed; the method of Newton, as delivered by
himself, is encumbered with difficulties and objections. Among
the ablest admirers of Newton, there have been obstinate dis
putes respecting his explanation of his method of " prime and
ultimate ratios."
The so-called "method of limits" is frequently attributed
to Newton, but the pure method of limits was never adopted
by Mm as his method of constructing the calculus. All he
did was to establish in his Principia certain principles which
are applicable to that method, but which he used for a different
purpose. The first lemma of the first book has been made the
foundation of the method of limits :
"Quantities and the ratios of quantities, which in any finite
time converge continually to equality, and before the end of
that time approach nearer the one to the other than by any
given difference, become ultimately equal."
In this, as well as in the lemmas following this, there are
obscurities and difficulties. Newton appears to teach that a
variable quantity and its limit will ultimately coincide and be
equal. But it is now generally agreed that in the clearest
statements which have been made of the theory of limits, the
variable does not actually reach its limit, though the variable
may approach it as near as we please.
The full title of Newton s Principia is Philosophic Natura-
lis Principia Mathematica. It was printed in 1687 under
the direction, and at the expense, of Dr. Edmund Halley.
A second edition was brought out in 1713 with many altera
tions and improvements, and accompanied by a preface from
NEWTON TO EULEB. 218
Mr. Cotes. It was sold out in a few months, but a pirated
edition published in Amsterdam supplied the demand. 34 The
third and last edition which appeared in England during
Newton s lifetime was published in 1726 by Henry Pemberton.
The Principia consists of three books, of which the first two,
constituting the great bulk of the work, treat of the mathe
matical principles of natural philosophy, namely, the laws and
conditions of motions and forces. In the third book is drawn
up the constitution of the universe as deduced from the fore
going principles. The great principle underlying this memor
able work is that of universal gravitation. The first book
was completed on April 28, 1686. After the remarkably short
period of three months, the second book was finished. The
third book is the result of the next nine or ten months
labours. It is only a sketch of a much more extended elabora
tion of the subject which he had planned, but which was never
brought to completion.
The law of gravitation is enunciated in the first book. Its
discovery envelops the name of Newton in a halo of perpetual
glory. The current version of the discovery is as follows : it
was conjectured by Hooke, Huygens, Halley, Wren, Newton,
and others, that, if Kepler s third law was true (its absolute
accuracy was doubted at that time), then the attraction
between the earth and other members of the solar system
varied inversely as the square of the distance. But the proof
of the truth or falsity of the guess was wanting. In 1666
Newton reasoned, in substance, that if g represent the acceler
ation of gravity on the surface of the earth, r be the earth s
radius, R the distance of the moon from the earth, T the time
of lunar revolution, and a a degree at the equator, then, if the-
law is true,
214 A HISTORY OF MATHEMATICS.
The data at Newton s command gave E = 60.4 r, T = 2,360,628
seconds, but a only 60 instead of 69 J English, miles. This
wrong value of a rendered the calculated value of g smaller
than its true value, as known from actual measurement. It
looked as though the law of inverse squares were not the true
law, and Newton laid the calculation aside: In 1684 he casu
ally ascertained at a meeting of the Royal Society that Jean
Picard had measured an arc of the meridian, and obtained a
more accurate value for the earth s radius. Taking the cor
rected value for a, he found a figure for g which corresponded
to the known value. Thus the law of inverse squares was
verified. In a scholium in the Prmcipm, Newton acknowl
edged his indebtedness to Huygens for the laws on centrifugal
force employed in his calculation.
The perusal by the astronomer Adams of a great mass of
unpublished letters and manuscripts of Newton forming the
Portsmouth collection (which -remained private property
until 1872, when its owner placed it in the hands of the
University of Cambridge) seems to indicate that the difficul
ties encountered by Newton in the above calculation were of a
different nature. According to Adams, Newton s numerical
verification was fairly complete in 1666, but Newton had not
been able to determine what the attraction of a spherical shell
upon an external point would be. His letters to Halley show
that he did not suppose the earth to attract as though all its
mass were concentrated into a point at the centre. He could
not have asserted, therefore, that the assumed law of gravity
was verified by the figures, though for long distances he might
have claimed that it yielded close approximations. When
Halley visited Newton in 1684, he requested Newton to deter
mine what the orbit of a planet would be if the law of attrac
tion were that of inverse squares. Newton had solved a
similar problem for Hooke in 1679, and replied at once that it
NEWTON TO EULER. 215
was an ellipse. After Halley s visit, Newton, with Picard s
new value for the earth s radius, reviewed his early calcula
tion, and was able to show that if the distances between the
bodies in the solar system were so great that the bodies might
be considered as points, then their motions were in accordance
with the assumed law of gravitation. In 1685 he completed
his discovery by showing that a sphere whose density at any
point depends only on the distance from the centre attracts
an external point as though its whole mass were concentrated
at the centre. 34
Newton s unpublished manuscripts in the Portsmouth col
lection show that he had worked out, by means of fluxions and
fluents, his lunar calculations to a higher degree of approxima
tion than that given in the Principia y but that he was unable
to interpret his results geometrically. The papers in that col
lection throw light upon the mode by which Newton arrived
at some of the results in the Principia, as, for instance, the
famous construction in Book II., Prop. 25, which is unproved
in the Principia, but is demonstrated by him twice in a draft
of a letter to David Gregory, of Oxford. 34
It is chiefly upon the Principia that the fame of Newton
rests. Brewster calls it "the brightest page in the records of
human reason." Let us listen, for a moment, to the comments
of Laplace, the foremost among those followers of Newton who
grappled with the subtle problems of the motions of planets
under the influence of gravitation : "Newton has well estab
lished the existence of the principle which he had the merit
of discovering, but the development of its consequences and
advantages has been the work of the successors of this great
mathematician. The imperfection of the infinitesimal calcu
lus, when first discovered, did not allow him completely to
resolve the difficult problems which the theory of the universe
offers ; and he was oftentimes forced to give mere hints, which
216 A HISTORY OF MATHEMATICS.
were always uncertain till confirmed by rigorous analysis.
Notwithstanding these unavoidable defects, the importance
and the generality of his discoveries respecting the system of
the universe, and the most interesting points of natural phi
losophy, the great number of profound and original views,
which have been the origin of the most brilliant discoveries of
the mathematicians of the last century, which were all pre
sented with much elegance, will insure to the Principia a last
ing pre-eminence over all other productions of the human
mind."
Newton s Arithmetica Uhiversalis, consisting of algebraical
lectures delivered by him during the first nine years he was
professor at Cambridge, were published in 1707, or more than
thirty years after they were written. This work was pub
lished by Mr. Whiston. We are not accurately informed how
Mr. Whiston came in possession of it, but according to some
authorities its publication was a breach of confidence on his
part.
The AritJimetica Uhiversalis contains new and important
results on the theory of equations. His theorem on the
sums of powers of roots is well known. Newton showed
that in equations with real coefficients, imaginary roots always
occur in pairs. His inventive genius is grandly displayed
in his rule for determining the inferior limit of the number
of imaginary roots, and the superior limits for the number
of positive and negative roots. Thoxigh less expeditious than
Descartes 7 , Newton s rule always gives as close, and generally
closer, limits to the number of positive and negative roots.
Newton did not prove his rule. It awaited demonstration
for a century and a half, until, at last, Sylvester established
a remarkable general theorem which includes Newton s rule
as a special case.
The treatise on Method of Fluxions contains Newton s method
NEWTON TO EULER. 217
of approximating to the roots of numerical equations. This
is simply the method of Vieta improved. The same treatise
contains " Newton s parallelogram/ 5 which enabled Mm, in an
equation, f(x } y) = 0, to find a series in powers of x equal to
the variable y. The great utility of this rule lay in its deter
mining the form of the series ; for, as soon as the law was
known by which the exponents in the series vary, then the
expansion could be effected by the method of indeterminate
coefficients. The rule is still used in determining the infinite
branches to curves, or their figure at multiple points. Newton
gave no proof for it, nor any clue as to how he discovered it.
The proof was supplied half a century later, by Kaestner and
Cramer, independently. 37
In 1704 was published, as an appendix to the OpticTcs, the
Enumeratio linearum tertii ordinis, which contains theorems
on the theory of curves. Newton divides cubics into seventy-
two species, arranged in larger groups, for which his com
mentators have supplied the names " genera " and " classes,"
recognising fourteen of the former and seven (or four) of the
latter. He overlooked six species demanded by his principles
of classification, and afterwards added by Stirling, Murdoch,
and Cramer. He enunciates the remarkable theorem that the
five species which he names "divergent parabolas" give by
their projection every cubic curve whatever. As a rule, the
tract contains no proofs. It has been the subject of frequent
conjecture how Newton deduced his results. Eecently we have
gotten at the facts, since much of the analysis used by Newton
and a few additional theorems have been discovered among the
Portsmouth papers. An account of the four holograph man
uscripts on this subject has been published by W. W. Eouse
Ball, in the Transactions of the London Mathematical Society
(vol. xx., pp. 104-143). It is interesting to observe how
Newton begins Ms research on the classification of cubic
218 A HISTOEY OF MATHEMATICS.
curves by the algebraic method, but, finding it laborious,
attacks the problem geometrically, and afterwards returns
again to analysis. 36
Space does not permit us to do more than merely mention
Newton s prolonged researches in other departments of science.
He conducted a long series of experiments in optics and is the
author of the corpuscular theory of light. The last of a
number of papers on optics, which he contributed to the Boyal
Society, 1687, elaborates the theory of " fits." He explained
the decomposition of light and the theory of the rainbow. By
him were invented the reflecting telescope and the sextant
(afterwards re-discovered by Thomas Godfrey of Philadelphia 2
and by John Hadley) . He deduced a theoretical expression
for the velocity of sound in air, engaged in experiments on
chemistry, elasticity, magnetism, and the law of cooling, and
entered upon geological speculations.
During the two years following the close of 1692, Newton
suffered from insomnia and nervous irritability. Some thought
that he laboured under temporary mental aberration. Though
he recovered his tranquillity and strength of mind, the time
of great discoveries was over ; he would study out questions
propounded to him, but no longer did he by his own accord
enter upon new fields of research. The most noted investi
gation after his sickness was the testing of his lunar theory
by the observations of Flamsteed, the astronomer royal. In
1695 he was appointed warden, and in 1699 master, of the
mint, which office he held until his death. His body was
interred in "Westminster Abbey, where in 1731 a magnificent
monument was erected, bearing an inscription ending with,
"Sibi gratulentur mortales tale tantumque exstitisse humani
generis decus." It is not true that the Binomial Theorem is
also engraved on it.
We pass to Leibniz, the second and independent inventor
NEWTON TO BULEB. 219
of the calculus. Gottfried Wilhelm Leibniz (1646-1716) was
born in Leipzig. No period in the history of any civilised
nation could have been less favourable for literary and scientific
pursuits than the middle of the seventeenth century in Ger
many. Yet circumstances seem to have happily combined
to bestow on the youthful genius an education hardly other
wise obtainable during this darkest period of German history.
He was brought early in contact with the best of the culture
then existing. In his fifteenth year he entered the University
of Leipzig. Though law was Ms principal study, he applied
himself with great diligence to every branch of knowledge.
Instruction in German universities was then very low. The
higher mathematics was not taught at all. We are told that
a certain John Kuhn lectured on Euclid s Elements, but that
his lectures were so obscure that none except Leibniz could
understand them. Later on, Leibniz attended, for a half-year,
at Jena, the lectures of Erhard Weigel, a philosopher and
mathematician of local reputation. In 1666 Leibniz published
a treatise, De Arte Combinatoria, in which he does not pass
beyond the rudiments of mathematics. Other theses written
by him at this time were metaphysical and juristical in
character. A fortunate circumstance led Leibniz abroad. In
1672 he was sent by Baron Boineburg on a political mission
to Paris. He there formed the acquaintance of the most
distinguished men of the age. Among these was Huygens,
who presented a copy of his work on the oscillation of the
pendulum to Leibniz, and first led the gifted young German
to the study of higher mathematics. In 1673 Leibniz went
to London, and remained there from January till March. He
there became incidentally acquainted with the mathematician
Pell, to whom he explained- a method he had found on the
summation of series of numbers by their differences. Pell
told him that a similar formula had been published by Mouton
220 A HISTORY OF MATHEMATICS,
as early as 1670, and then called his attention to Mercator s
work on the rectification of the parabola. While in London,
Leibniz exhibited to the Koyal Society his arithmetical ma
chine;* which was similar to Pascal s, but more efficient and
perfect. After his return to Paris, he had the leisure to study
mathematics more systematically. With indomitable energy
he set about removing his ignorance of higher mathematics.
Huygens was his principal master. He studied the geometric
works of Descartes, Honorarius Fabri, Gregory St. Vincent,
and Pascal. A careful study of infinite series led him to the
discovery of the following expression for the ratio of the
circumference to the diameter of the circle :
This elegant series was found in the same way as Mercator s
on the hyperbola. Huygens was highly pleased with it and
urged him on to new investigations. Leibniz entered into a
detailed study of the quadrature of curves and thereby became
intimately acquainted with the higher mathematics. Among
the papers of Leibniz is still found a manuscript on quadra
tures, written before he left Paris in 1676, but which was
never printed by him. The more important parts of it were
embodied in articles published later in the Acta Eruditorum.
In the study of Cartesian geometry the attention of Leibniz
was drawn early to the direct and inverse problems of tan
gents. The direct problem had been solved by Descartes for
the simplest curves only; while the inverse had completely
transcended the power of his analysis. Leibniz investigated
both problems for any curve ; he constructed what he called
tne triangulum characteristicum an infinitely small triangle
between the infinitely small part of the curve coinciding with
the tangent, and the differences of the ordinates and abscissas.
NEWTON TO EULER. 221
A curve is liere considered to be a polygon. The trianyulum
characteristicum is similar to the triangle formed by the tan
gent, the ordinate of the point of contact, and the sub-
tangent, as well as to that between the ordinate, normal,
and sub-normal. It was first employed by Barrow in Eng
land, but appears to have been reinvented by Leibniz. [From
it Leibniz observed the connection existing between the direct
and inverse problems of tangents. He saw also that the latter
could be carried back to the quadrature of curves. All these
results are contained in a manuscript of Leibniz, written in
1673. One mode used by him in effecting quadratures was
as follows : The rectangle formed by a sub-tangent p and
an element a (i.e. infinitely small part of the abscissa) is
equal to the rectangle formed by the ordinate y and the ele
ment I of that ordinate; or in symbols, pa,=*yL But the
summation of these rectangles from zero on gives a right
triangle equal to half the square of the ordinate. Thus,
using Cavalieri s notation, he gets
7,2
omn. pa = omn. yl = &- (omn. meaning omnia, all) .
Jj
But y = omn. I ; hence
omn. omn. I - =
a
I omn. Z 2
This equation is especially interesting, since it is here that
Leibniz first introduces a new notation. He says: "It will
be useful to write \ for omn., as f I for omn. I, that is, the
sum of the Z s " ; he then writes the equation thus :
a,
Erom, this he deduced the simplest integrals, such as
222 A HISTOKY OF MATHEMATICS.
Since tlie symbol of summation J raises the dimensions, he
concluded that the opposite calculus, or that of differences
d, would lower them. Thus, if \ I = TO, then I = ^. The
J d
symbol d was at first placed by Leibniz in the denominator,
because the lowering of the power of a term was brought
about in ordinary calculation by division. The manuscript
giving the above is dated October 29th, 1675. 39 This, then,
was the memorable day on which the notation of the new
calculus came to be, a notation which contributed enor
mously to the rapid growth and perfect development of the
calculus.
Leibniz proceeded to apply his new calculus to the solution
of certain problems then grouped together under the name
of the Inverse Problems of Tangents. He found the cubical
parabola to be the solution to the following: To find the
curve in which the sub-normal is reciprocally proportional
to the ordinate. The correctness of his solution was tested
by him by applying to the result Sluze s method of tangents
and reasoning backwards to the original supposition. In the
solution of the third problem he changes his notation from
to the now usual notation dx. It is worthy of remark
d
that in these investigations, Leibniz nowhere explains the
significance of dx and dy, except at one place in a marginal
note: "Idem est dx et -, id est, differentia inter duas x
d
proximas." E"or does he use the term differential, but
always difference. Not till ten years later, in the Acta
Eruditorum, did he give further explanations of these sym
bols. What he aimed at principally was to determine the
change an expression undergoes when the symbol f or d is
placed before it. It may be a consolation to students wres
tling with the elements of the differential calculus to know
that it required Leibniz considerable thought and atten-
KEWTCXN" TO ETJLEK. 223
tion 39 to determine whether dxdy is the same as d(xy}, and
the same as d~. After considering these questions at
dy y
the close of one of his manuscripts, he concluded that the
expressions were not the same, though he could not give the
true value for each. Ten days later, in a manuscript dated
November 21, 1675, he found the equation ydx^dHcy xdy,
giving an expression for d(xy^), which he observed to be true
for all curves. He succeeded also in eliminating dx from
a differential equation, so that it contained only dy, and
thereby led to the solution of the problem under considera
tion. "Behold, a most elegant way by which the problems
of the inverse methods of tangents are solved, or at least
are reduced to quadratures ! " Thus he saw clearly that the
inverse problems of tangents could be solved by quadratures,
or, in other words, by the integral calculus. In course of a
half-year he discovered that the direct problem of tangents,
too, yielded to the power of his new calculus, and that thereby
a more general solution than that of Descartes could be
obtained. He succeeded in solving all the special problems
of this kind, which had been left unsolved by Descartes.
Of these we mention only the celebrated problem proposed
to Descartes by De Beaune, viz. to find the curve whose
ordinate is to its sub-tangent as a given line is to that part
of the ordinate which lies between the curve and a line drawn
from the vertex of the curve at a given inclination to the axis.
Such was, in brief, the progress in the evolution of the new
calculus made by Leibniz during his stay in Paris. Before
his depasture, in October, 1676, he found himself in possession
of the most elementary rules and formulae of the infinitesimal
calculus.
From Paris, Leibniz returned to Hanover by way of London
and Amsterdam. In London he met Collins, who showed him
224 A HISTORY OF MATHEMATICS.
a part of Ms scientific correspondence. Of this we shall speak
later. In Amsterdam he discussed mathematics with Sluze,
and became satisfied that his own method of constructing
tangents not only accomplished all that Sluze s did, but even
more, since it could be extended to three variables, by which
tangent planes to surfaces could be found; and especially,
since neither irrationals nor fractions prevented the immediate
application of his method.
In a paper of July 11, 1677, Leibniz gave correct rules for
the differentiation of sums, products, quotients, powers, and
roots. He had given the differentials of a few negative and
fractional powers, as early as November, 1676, but had made
some mistakes. For d Vcc he had given the erroneous value
-r, and in another place the value 4ar^ : for d-z occurs in
V 2
one place the wrong value, -, while a few lines lower is
3 . ^
given j its correct value.
In 1682 was founded in Berlin the Acta jEruditorum, a
journal usually known by the name of Leipzig Acts. It
was a partial imitation of the French Journal des JSavans
(founded in 1665), and the literary and scientific review
published in Germany. Leibniz was a frequent contributor.
Tschirnhaus, who had studied mathematics in Paris with
Leibniz, and who was familiar with the new analysis of
Leibniz, published in the Acta Eroditorum a paper on quad
ratures, which consists principally of subject-matter com
municated by Leibniz to Tschirnhaus during a controversy
which they had had on this subject. Fearing that Tschirnhaus
might claim as his own and publish the notation and rules of
the differential calculus, Leibniz decided, at last, to make
public the fruits of his inventions. In 1684, or nine years
after the new calculus first dawned upon the mind of Leibniz,
and nineteen years after Newton first worked at fluxions,
HEWTON TO ETJLEB. 225
and three years before the publication of Newton s Principia,,
Leibniz published, in the Leipzig Acts, Ms first paper on the
differential calculus. He was unwilling to give to the world
-all his treasures, but chose those parts of his work which were
most abstruse and least perspicuous. This epoch-making paper
of only six pages bears the title : f( ISTova methodus pro maxiinis
et minimis, itemque tangentibus, quae nee fraetas nee irra-
tionales quantitates moratur, et singulars pro illis calculi
genus. 5 The rules of calculation are briefly stated without
proof, and the meaning of dx and dy is not made clear. It has
been inferred from this that Leibniz himself had no definite
and settled ideas on this subject. Are dy and dx finite or
infinitesimal quantities ? At first they appear, indeed, to have
been taken as finite, when he says : " We now call any line
selected at random dx, then we designate the line which is to
dx as y is to the sub-tangent, by dy, which is the difference of
?/." Leibniz then ascertains, by his calculus, in what way a
ray of light passing through two differently refracting media,
can travel easiest from one point to another ; and then closes
his article by giving his solution, in a few words, of De
Beaune s problem. Two years later (1686) Leibniz published
in the Acta Eruditorwm, a paper containing the rudiments of
the integral calculus. The quantities dx and dy are there
treated as infinitely small. He showed that by the use of his
notation, the properties of curves could be fully expressed by
equations. Thus the equation
, r &B
y = -V2x x* + \ .
* J V205
characterises the cycloid. 88
The great invention of Leibniz, now made public by his
articles in the Leipzig Acts, made little impression upon the
mass of mathematicians. In Germany no one comprehended
226 A HISTORY OF MATHEMATICS.
the new calculus except Tschirnhaus, who remained indif
ferent to it. The author s statements were too short and suc
cinct to make the calculus generally understood. The first
to recognise its importance and to take up the study of it
were two foreigners, the Scotchman TJiomas Craige, and
the Swiss James Bernoulli. The latter wrote Leibniz a
letter in 1687, wishing to be initiated into the mysteries
of the new analysis. Leibniz was then travelling abroad,
so that this letter remained unanswered till 1^)0. James
Bernoulli succeeded, meanwhile, by close application; in un
covering the secrets of the differential calculus without assist
ance. He and his brother John proved to be mathematicians
of exceptional power. They applied themselves to the new
science with a success and to an extent which made Leibniz
declare that it was as much theirs as his. Leibniz carried on
an extensive correspondence with them, as well as with other
mathematicians. In a letter to John Bernoulli he suggests,
among other things, that the integral calculus be improved by
reducing integrals back to certain fundamental irreducible
forms. The integration of logarithmic expressions was then
studied. The writings of Leibniz contain many innovations,
and anticipations of since prominent methods. Thus he made
use of variable parameters, laid the foundation of analysis in
situ, introduced the first notion of determinants in his effort
to simplify the expression arising in the elimination of the
unknown quantities from a set of linear equations. He
resorted to the device of breaking up certain fractions into
the sum of other fractions for the purpose of easier integration ;
he explicitly assumed the principle of continuity ; he gave the
first instance of a " singular solution," and laid the foundation
to the theory of envelopes in two papers, one of which contains
for the first time the terms co-ordinate and axes of co-ordinates.
He wrote on osculating curves, but his paper contained the
NEWTON TO EULEB. 227
error (pointed out by John Bernoulli, but not admitted by
Mm) that an osculating circle will necessarily cut a curve in
four consecutive points. Well known is his theorem on the
nth differential coefficient of the product of two functions of a
variable. Of his many papers on mechanics, some are valuable,
while others contain grave errors.
Before tracing the further development of the calculus we
shall sketch the history of that long and bitter controversy
between English and Continental mathematicians on the inven
tion of the calculus. The question was, did Leibniz invent it
independently of JSTewton, or was he a plagiarist ?
We must begin with the early correspondence between the
parties appearing in this dispute. Fewfcon had begun using
his notation of fluxions in 1666, 41 In 1669 Barrow sent Collins
Newton s tract, De Analysi per Equationes, etc.
The first visit of Leibniz to London extended from the llth
of January until March, 1673. He was in the habit of com
mitting to writing important scientific communications received
from others. In 1890 Gerhardt discovered in the royal library
at Hanover a sheet of manuscript with notes taken by Leibniz
during this journey. 40 They are headed " Observata Philoso-
phica in itinere Anglicano sub initium anni 1673." The sheet
is divided by horizontal lines into sections. The sections
given to Chymica, Mechanica, Magnetica, Botaniea, Anatomica,
Medica, Miscellanea, contain extensive memoranda, while those
devoted to mathematics have very few notes. Under G-eo-
metrica he says only this : " Tangentes omnium figurarum.
Figurarum geometricarum explicatio per motum puncti in
moto lati." We suspect from this that Leibniz had read
Barrow s lectures. Newton is referred to only under Optica.
Evidently Leibniz did not obtain a knowledge of fluxions
during this visit to London, nor is it claimed that he did by
his opponents.
228 A HISTOBY OF MATHEMATICS.
Various -letters of Newton, Collins, and others, up to the
beginning of 1676, state that Newton invented a method by
which tangents could be drawn without the necessity of freeing
their equations from irrational terms. Leibniz announced in
1674 to Oldenburg, then secretary of the Eoyal Society, that
he possessed very general analytical methods, by which he had
found theorems of great importance on the quadrature of the
circle by means of series. In answer, Oldenburg stated
Newton and James Gregory had also discovered methods of
quadratures, which extended to the circle. Leibniz desired to
have these methods communicated to him; and Newton, at
the request of Oldenburg and Collins, wrote to the former the
celebrated letters of June 13 and October 24, 1676. The first
contained the Binomial Theorem and a variety of other mat-
ters relating to infinite series and quadratures; but nothing
directly on the method of fluxions. Leibniz in reply speaks
in the highest terms of what Newton had done, and requests
further explanation. Newton in his second letter just men
tioned explains the way in which he found the Binomial
Theorem, and also communicates his method of fluxions and
fluents in form of an anagram in which all the letters in the
sentence communicated were placed in alphabetical order.
Thus Newton says that his method of drawing tangents was
Gaccdce IBejf 7i 31 9n 40 4qrr 4s 9t 12vx.
The sentence was, "Data sequatione quotcunque fluentes
quantitates involvente fluxiones invenire, et vice versa."
("Having any given equation involving never so many flowing
quantities, to find the fluxions, and vice versa.") Surely this
anagram afforded no hint. Leibniz wrote a reply to Collins,
in which, without any desire of concealment, he explained the
principle, notation, and the use of the differential calculus.
The death of Oldenburg brought this correspondence to a
NEWTO2ST TO EULEB. 229
close. Nothing material happened till 1684, when Leibniz
published his first paper on the differential calculus in the
Leipzig Acts, so that while Newton s claim to the priority of
invention must be admitted by all, it must also be granted
that Leibniz was the first to give the full benefit of the calcu
lus to the world. Thus, while Newton s invention remained a
secret, communicated only to a few friends, the calculus of
Leibniz was spreading over the Continent. No rivalry or
hostility existed, as yet, between the illustrious scientists.
Newton expressed a very favourable opinion of Leibniz s
inventions, known to him through the above correspondence
with Oldenburg, in the following celebrated scholium (Prmci-
pia, first edition, 1687, Book II., Prop. 7, scholium) :
" In letters which went between me and that most excellent
geometer, G. G-. Leibniz, ten years ago, when I signified that
I was in the knowledge of a method of determining maxima
and minima, of drawing tangents, and the like, and when I
concealed it in transposed letters involving this sentence (Data
gequatione, etc., above cited), that most distinguished man
wrote back that he had also fallen upon a method of the same
kind, and communicated his method, which hardly differed
from mine, except in his forms of words and symbols."
As regards this passage, we shall see that Newton was after
wards weak enough, as De Morgan says : " First, to deny the
plain and obvious meaning, and secondly, to omit it entirely
from the third edition of the Principia.^ On the Continent,
great progress was made in the calculus by Leibniz and his
coadjutors, the brothers James and John Bernoulli, and
Marquis de PHospital. In 1695 Wallis informed Newton by
letter that " he had heard that his notions of fluxions passed
in Holland with great applause by the name of Leibniz s
Calculus Differentialis. " Accordingly Wallis stated in the
preface to a volume of his works that the calculus differen-
230 A HISTOKY OF MATHEMATICS.
tialis was Newton s method of fluxions which had been
communicated to Leibniz in the Oldenburg letters. A review
of Wallis works, in the Leipzig Acts for 1696, reminded the
reader of Newton s own admission in the scholium above
cited.
For fifteen years Leibniz had enjoyed unchallenged the
honour of being the inventor of his calculus. But in 1699 Fato
de Duillier, a Swiss, who had settled in England, stated in a
mathematical paper, presented to the Royal Society, his con
viction that Newton was the first inventor; adding that,
whether Leibniz, the second inventor, had borrowed anything
from the other, he would leave to the judgment of those who
had seen the letters and manuscripts of Newton. This was
the first distinct insinuation of plagiarism. It would seem that
the English mathematicians had for some time been cherishing
suspicions unfavourable to Leibniz. A feeling had doubtless
long prevailed that Leibniz, during his second visit to London
in 1676, had or might have seen among the papers of Collins
Newton s Analysis per cequationes, etc., which contained appli
cations of the fluxionary method, but no systematic develop
ment or explanation of it. Leibniz certainly did see at least
part of this tract. During the week spent in London, he took
note of whatever interested him among the letters and papers
of Collins. His memoranda discovered by Gerhardt in 1849 in
the Hanover library fill two sheets. 40 The one bearing on our
question is headed "Excerpta ex tractatu Newtoni Msc. de
Analysi per sequationes numero terminorum infinitas." The
notes are very brief, excepting those De Resolutions cequa-
tionum qffectarum, of which there is an almost complete copy.
This part was evidently new to him. If he examined
Newton s entire tract, the other parts did not particularly
impress him. From it he seems to have gained nothing per
taining to the infinitesimal calculus. By the previous intro-
HEWTON TO EULEB. 231
duction of his own algorithm he had made greater progress
than by what eaine to his knowledge in London. Nothing
mathematical that he had received engaged his thoughts in
the immediate future, for on his way back to Holland he com
posed a lengthy dialogue on mechanical subjects.
Duillier s insinuations lighted up a name of discord which a
whole century was hardly sufficient to extinguish. Leibniz,
who had never contested the priority of Newton s discovery,
and who appeared to be quite satisfied with Newton s admis
sion in his scholium, now appears for the first time in the
controversy. He made an animated reply in the Leipzig Acts,
and complained to the Royal Society of the injustice done him.
Here the affair rested for some time. In the Quadrature of
Curves, published 1704, for the first time, a formal exposition
of the method and notation of fluxions was made public. In
1T05 appeared an unfavourable review of this in the Leipzig
Acts, stating that Newton uses and always has used fluxions
for the differences of Leibniz. This was considered by New
ton s friends an imputation of plagiarism on the part of their
chief, but this interpretation was always strenuously resisted
]by Leibniz. ELeill, professor of astronomy at Oxford, under
took with more zeal than judgment the defence of Newton.
In a paper inserted in the Philosophical Transactions of 1708,
he claimed that Newton was the first inventor of fluxions and
" that the same calculus was afterward published by Leibniz,
the name and the mode of notation being changed." Leibniz
complained to the secretary of the Royal Society of bad treat
ment and requested the interference of that body to induce
Keill to disavow the intention of imputing fraud. Keill was
not made to retract his accusation; on the contrary, was
authorised by Newton and the Eoyal Society to explain and
defend his statement. This he did in a long letter. Leibniz
thereupon complained that the charge was now more open than
232 A HISTOEY OF MATHEMATICS.
before, and appealed for justice to the Eoyal Society and to
Newton himself. The Eoyal Society, thus appealed to as a
judge, appointed a committee which collected and reported
upon a large mass of documents mostly letters from and to
Newton, Leibniz, Wallis, Collins, etc. This report, called the
Commerdum Hpistolicum, appeared in the year 1712 and again
in 1725, with a Eecensio prefixed, and additional notes by Keill.
The final conclusion in the Commerdum Epistolicum was
that Newton was the first inventor. But this was not to the
point. The question was not whether Newton was the first
inventor, but whether Leibniz had stolen the method. The
committee had not formally ventured to assert their belief
that Leibniz was a plagiarist. Yet there runs throughout the
document a desire of proving Leibniz guilty of more than
they meant positively to affirm. Leibniz protested only in
private letters against the proceeding of the Eoyal Society,
declaring that he would not answer an argument so weak.
John Bernoulli, in a letter to Leibniz, which was published
later in an anonymous tract, is as decidedly unfair towards
Newton as the friends of the latter had been towards Leibniz.
Keill replied, and then Newton and Leibniz appear as mutual
accusers in several letters addressed to third parties. In a
letter to Conti, April 9, 1716, Leibniz again reminded Newton
of the admission he had made in the scholium, which he was
now desirous of disavowing; Leibniz also states that he
always believed Newton, but that, seeing him connive at
accusations which he must have known to be false, it was
natural that he (Leibniz) should begin to doubt. Newton
did not reply to this letter, but circulated some remarks among
his friends which he published immediately after hearing
of the death of Leibniz, November 14, 1716. This paper
of Newton gives the following explanation pertaining to the
scholium in question: "He [Leibniz] pretends that in my
NEWTON TO BTJLBE. 233
book of principles I allowed him the invention of the calculus
diff erentialis, independently of my own ; and that to attribute
this invention to . myself is contrary to my knowledge there
avowpd. But in the paragraph there referred unto I do not
find one word to this purpose." In the third edition of the
Principm, 1725, Newton omitted the scholium and substituted
in its place another, in which the name of Leibniz does not
appear.
National pride and party feeling long prevented the adoption
of impartial opinions in England, but now it is generally ad
mitted by nearly all familiar with the matter, that Leibniz
really was an independent inventor. Perhaps the most tell
ing evidence to show that Leibniz was an independent inven
tor is found in the study of his mathematical papers (collected
and edited by C. I. Gerhardt, in six volumes, Berlin, 1849-
1860), which point out a gradual and natural evolution of the
rules of the calculus in his own mind. " There was through
out the whole dispute," says De Morgan, " a confusion between
the knowledge of fluxions or differentials and that of a calcu
lus of fluxions or differentials ; that is, a digested method with
general rules."
This controversy is to be regretted on account of the long
and bitter alienation which it produced between English and
Continental mathematicians. It stopped almost completely
all interchange of ideas on scientific subjects. The English
adhered closely to Newton s methods and, until about 1820,
remained, in most cases, ignorant of the brilliant mathematical
discoveries that were being made on the Continent. The loss
in point of scientific advantage was almost entirely on the
side of Britain. The only way in which this dispute may be
said, in a small measure, to have furthered the progress of
mathematics, is through the challenge problems by which
each side attempted to annoy its adversaries.
284 A HISTORY OF MATHEMATICS.
The recurring practice of issuing challenge problems was
inaugurated at this time by Leibniz. They were, at first, not
intended as defiances, but merely as exercises in the new cal
culus. Such was the problem of the isochronous curve (to
find the curve along which a body falls with uniform velocity),
proposed by him to the Cartesians in 1687, and solved by
James Bernoulli, himself, and John Bernoulli. James Ber
noulli proposed in the Leipzig Journal the question to find the
curve (the catenary) formed by a chain of uniform weight
suspended freely from its ends. It was resolved by Huygens,
Leibniz, and himself. In 1697 John Bernoulli challenged the
best mathematicians in Europe to solve the difficult problem,
to find the curve (the cycloid) along which a body falls from
one point to another in the shortest possible time. Leibniz
solved it the day he received it. Newton, de PHospital, and
the two Bernoullis gave solutions. Newton s appeared anony
mously in the Philosophical Transactions, but John Bernoulli
recognised in it his powerful mind, "tanquam," he says, "ex
ungne leonein." The problem of orthogonal trajectories (a
system of curves described by a known law being given, to
describe a curve which shall cut them all at right angles) had
been long proposed in the Acta Eruditorum, but failed at
first to receive much attention. It was again proposed in
1716 by Leibniz, to feel the pulse of the English mathema
ticians.
This may be considered as the first defiance problem pro
fessedly aimed at the English. Newton solved it the same
evening on which it was delivered to him, although he was
much fatigued by the day s work at the mint. His solution,
as published, was a general plan of an investigation rather
than an actual solution, and was, on that account, criticised by
Bernoulli as being of no value. Brook Taylor undertook the
defence of it, but ended by using very reprehensible language.
NEWTON TO ETJLEB. 235
Bernonlli was not to be outdone in incivility, and made a
bitter reply. Not long afterwards Taylor sent an open de
fiance to Continental mathematicians of a problem on the
integration of a fluxion of complicated form which was known
to very few geometers in England and supposed to be beyond
the power of their adversaries. The selection was injudicious,
for Bernoulli had long before explained the method of this
and similar integrations. It served only to display the skill
and augment the triumph of the followers of Leibniz. The
last and most unskilful challenge was by John Keill. The
problem was to find the path of a projectile in a medium
which resists proportionally to the square of the velocity.
Without first making sure that he himself could solve it,
Keill boldly challenged Bernoulli to produce a solution. The
latter resolved the question in very short time, not only for a
resistance proportional to the square, but to any power of the
velocity. Suspecting the weakness of the adversary, he re
peatedly offered to send his solution to a confidential person
in London, provided Keill would do the same. Keill never
made a reply, and Bernoulli abused him and cruelly exulted
over him. 26
The explanations of the fundamental principles of the cal
culus, as given by Newton and Leibniz, lacked clearness and
rigour. For that reason it met with opposition from several
quarters. In 1694 Bernard Nieuwentyt of Holland denied
the existence of differentials of higher orders and objected to
the practice of neglecting infinitely small quantities. These
objections Leibniz was not able to meet satisfactorily. In his
reply he said the value of -^ in geometry could be expressed
as the ratio of finite quantities. In the interpretation of dx
and dy Leibniz vacillated. At one time they appear in his
writings as finite lines ; then they are called infinitely small
236 A HISTORY OF MATHEMATICS.
quantities, and again, quantitates inassignabiles, which spring
from quantitates assignabiles by the law of continuity. In this
last presentation Leibniz approached nearest to Newton.
In England the principles of fluxions were boldly attacked
by Bishop Berkeley, the eminent metaphysician, who argued
with great acuteness, contending, among other things, that
the fundamental idea of supposing a finite ratio to exist
between terms absolutely evanescent "the ghosts of de
parted quantities," as he called them was absurd and unin
telligible. The reply made by Jurin failed to remove all the
objections. Berkeley was the first to point out what was
again shown later by Lazare Garnet, that correct answers were
reached by a " compensation of errors." Berkeley s attack
was not devoid of good results, for it was the immediate cause
of the work on fluxions by Maclaurin. In France Michel
Rolle rejected the differential calculus and had a controversy
with Varignon on the subject.
Among the most vigorous promoters of the calculus on the
Continent were the Bernoullis. They and Euler made Basel
in Switzerland famous as the cradle of great mathematicians.
The family of Bernoullis furnished in course of a century
eight members who distinguished themselves in mathematics.
We subjoin the following genealogical table :
Nicolaus Bernoulli, the Father
Jacob, 1654-1705 Nicolaus Johann, 1667-1748
Nicolaus, 1687-1759 Nicolaus, 1695-1726
Daniel, 1700-1782
Johann, 1710-1790
Daniel Johann, 1744-1807 Jacob, 1758-1789
Most celebrated were the two brothers Jacob (James) and
Johann (John), and Daniel, the son of, John. James and
NEWTON TO BULEB. 237
John were staunch friends of Leibniz and worked hand in
hand with him. James Bernoulli (1654-1705) was born in
Basel. Becoming interested in the calculus, he mastered it
without aid from a teacher. From 1687 until his death he
occupied the mathematical chair at the University of Basel.
He was the first to give a solution to Leibniz s problem of the
isochronous curve. In his solution, published in the Acta
Eruditorum, 1690, we meet for the first time with the word
integral. Leibniz had called the integral calculus calculus
summatoriuS) but in 1696 the term calculus integralis was
agreed upon between Leibniz and John Bernoulli. James
proposed the problem of the catenary, then proved the correct
ness of Leibniz s construction of this curve, and solved the
more complicated problems, supposing the string to be (1) of
variable density, (2) extensible, (3) acted upon at each point
by a force directed to a fixed centre. Of these problems he
published answers without explanations, while his brother
John gave in addition their theory. He determined the shape
of the " elastic curve " formed by an elastic plate or rod fixed
at one end and bent by a weight applied to the other end ; of
the "lintearia," a flexible rectangular plate with two sides
fixed horizontally at the same height, filled with a liquid ; of
the ({ volaria," a rectangular sail filled with wind. He studied
the loxodromic and logarithmic spirals, in the last of which
he took particular delight from its remarkable property of
reproducing itself under a variety of conditions. Following
the example of Archimedes, he willed that the curve be en
graved upon his tombstone with the inscription " eadem mutata
resurgo." In 1696 he proposed the famous problem of isoper-
imetrical figures, and in 1701 published his own solution. He
wrote a work on Ars Conjectandi, which is a development of
the calculus of probabilities and contains the investigation
now called "Bernoulli s theorem " and the so-called "numbers
238 A HISTORY OF MATHEMATICS.
of Bernoulli," which are in fact (though not so considered by
Mm) the coefficients of in the expansion of (e x I)- 1 , of
nl
his collected works, in three volumes, one was printed in 1713,
the other two in 1744.
John Bernoulli (1667-1748) was initiated into mathematics
by his brother. He afterwards visited France, where he met
Malebranche, Cassini, De Lahire, Yarignon, and de PHospital.
For ten years he occupied the mathematical chair at Groningen
and then succeeded his brother at Basel. He was one of the
most enthusiastic teachers and most successful original inves
tigators of his time. He was a member of almost every learned
society in Europe. His controversies were almost as numerous
as his discoveries. He was ardent in his friendships, but
unfair, mean, and violent toward all who incurred his dislike
even his own brother and son. He had a bitter dispute
with James on the isoperimetrical problem. James convicted
him of several paralogisms. After his brother s death he
attempted to substitute a disguised solution of the former for
an incorrect one of his own. John admired the merits of
Leibniz and Euler, but was blind to those of Newton. He
immensely enriched the integral calculus by his labours.
Among his discoveries are the exponential calculus, the line
of swiftest descent, and its beautiful relation to the path
described by a ray passing through strata of variable density.
He treated trigonometry by the analytical method, studied
caustic curves and trajectories. Several times he was given
prizes by the Academy of Science in Paris.
Of his sons, Nicholas and Daniel were appointed professors
of mathematics at the same time in the Academy of St.
Petersburg. The former soon died in the prime of life; the
latter returned to Basel in 1733, where he assumed the chair
of experimental philosophy. His first mathematical publi-
TO EULEB. 239
cation was tlie solution of a differential equation proposed by
Baccati. He wrote a work on hydrodynamics. His investiga
tions on probability are remarkable for their boldness and
originality. He proposed the theory of moral expectation,
which he thought would give results more in accordance with
our ordinary notions than the theory of mathematical prob
ability. His " moral expectation " has become classic, but no
one ever makes use of it. He applies the theory of probability
to insurance ; to determine the mortality caused by small-pox
at various stages of life ; to determine the number of survivors
at a given age from a given number of births ; to determine
how much inoculation lengthens the average duration of life.
He showed how the differential calculus could be used in the
theory of probability. He and Euler enjoyed the honour of
having gained or shared no less than ten prizes from the
Academy of Sciences in Paris.
Jofcann Bernoulli (born 1710) succeeded his father in the
professorship of mathematics at Basel. He captured three
prizes (on the capstan, the propagation of light, and the
magnet) from the Academy of Sciences at Paris. Micolaus
Bernoulli (born 1687) held for a time the mathematical chair
at Padua which Galileo had once filled. Johami Bernoulli
(born 1744) at the age of nineteen was appointed astronomer
royal at Berlin, and afterwards director ,of the mathematical
department of the Academy. His brother Jacob took upon
himself the duties of the chair of experimental physics at
Basel, previously performed by his uncle Jacob, and later
was appointed mathematical professor in the Academy at St.
Petersburg.
Brief mention will now be made of some other mathemati
cians belonging to the period of Newton, Leibniz, and the
elder Bernoullis.
GuiUaume Francois Antoine 1 Hospital (1661-1704), a pupil
240 A HISTORY OF MATHEMATICS.
of John Bernoulli, has already been mentioned as taking
part in the challenges issued by Leibniz and the Bernoullis.
He helped powerfully in making the calculus of Leibniz better
known to the mass of mathematicians by the publication of a
treatise thereon in 1696. This contains for the first time the
method of finding the limiting value of a fraction whose two
terms tend toward zero at the same time .
Another zealous French advocate of the calculus was Pierre
Varignon (1654-1722). Joseph Saurin (1659-1737) solved the
delicate problem of how to determine the tangents at the
multiple points of algebraic curves. Francois Nicole (1683-
1758) in 1717 issued the first systematic treatise on finite
differences, in which he finds the sums of a considerable
number of interesting series. He wrote also on roulettes,
particularly spherical epicycloids, and their rectification. Also
interested in finite differences was Pierre Raymond de Montmort
(1678-1719). His chief writings, on the theory of probabil
ity, served to stimulate his more distinguished successor, De
Moivre. Jean Paul de Gua (1713-1785) gave the demonstration
of Descartes rule of signs, now given in books. This skilful
geometer wrote in 1740 a work on analytical geometry, the
object of which was to show that most investigations on curves
could be carried on with the analysis of Descartes quite as
easily as with the calculus. He shows how to find the tan
gents, asymptotes, and various singular points of curves of all
degrees, and proved by perspective that several of these points
can be at infinity. A mathematician who clung to the methods
of the ancients was Philippe de Lahire (1640-1718), a pupil of
Desargues. His work on conic sections is purely synthetic,
but differs from ancient treatises in deducing the properties of
conies from those of the circle in the same manner as did
Desargues and Pascal. His innovations stand in close relation
with modern synthetic geometry. He wrote on roulettes, on
NEWTON TO EULER. 241
graphical methods, epicycloids, conchoids, and on magic
squares. Michel Rolle (1652-1719) is the author of a theorem
named after him.
Of Italian mathematicians, Eiccati and Fagnano must not
remain unmentioned. Jacopo Francesco, Count Riccati (1676-
1754) is best known in connection with his problem, called
Eiccati ? s equation, published in the Acta Eruditorum in 1724.
He succeeded in integrating this differential equation for some
special cases. A geometrician of remarkable power was Giulio
Carlo, Count de Fagnano (1682-1766). He discovered the fol
lowing formula, 7r =2nog^_^ in which he anticipated Euler
-j- 1
in the use of imaginary exponents and logarithms. His studies
on the rectification of the ellipse and hyperbola are the start
ing-points of the theory of elliptic functions. He showed, for
instance, that two arcs of an ellipse can be found in an in
definite number of ways, whose difference is expressible by a
right line.
In Germany the only noted contemporary of Leibniz is
Ehrenfried Walter TscMrnhausen (1631-1708), who discovered
the caustic of reflection, experimented on metallic reflectors
and large burning-glasses, and gave us a method of transform
ing equations named after him. Believing that the most
simple methods (like those of the ancients) are the most
correct, he concluded that in the researches relating to the
properties of curves the calculus might as well be dispensed
with.
After the death of Leibniz there was in Germany not a
single mathematician of note. Christian Wolf (1679-1754),
professor at Halle, was ambitious to figure as successor of
Leibniz, but he " forced the ingenious ideas of Leibniz into a
pedantic scholasticism, and had the unenviable reputation of
having presented the elements of the arithmetic, algebra, and
24-2 A HISTOBY OF MATHEMATICS,
analysis developed since the time of the Renaissance in the
form of Euclid, of course only in outward form, for into the
spirit of them he was quite unable to penetrate." I6
The contemporaries and immediate successors of Newton in
Great Britain were men of no mean merit. We have refer
ence to Cotes, Taylor, Maclaurin, and Be Moivre. We are
told that at the death of Roger Cotes (1682-1716), Newton
exclaimed, " If Cotes had lived, we might have known some
thing." It was at the request of Dr. Bentley that Cotes
undertook the publication of the second edition of Newton s
Principia. His mathematical papers were published after his
death by Eobert Smith, his successor in the Plunibian pro
fessorship at Trinity College. The title of the work, Har-
monia Mensumrum, was suggested by the following theorem
contained in it : If on each radius vector, through a fixed point
0, there be taken a point It, such that the reciprocal of OR be
the arithmetic mean of the reciprocals of OE^ OE 2} OE n ,
then the locus of R will be a straight line. In this work
progress was made in the application of logarithms and the
properties of the circle to the calculus of fluents. To Cotes
we owe" a theorem in trigonometry which depends on the
forming of factors of x n 1. Chief among the admirers of
Newton were Taylor and Maclaurin. The quarrel between
English and Continental mathematicians caused them to work
quite independently of their great contemporaries across the
Channel.
Brook Taylor (1685-1731) was interested in many branches
of learning, and in the latter part of his life engaged mainly in
religious and philosophic speculations. His principal work,
Methodus incrementorum directa et inversa, London, 1715-1717,
added a new branch to mathematics, now called " finite differ
ences." He made many important applications of it, par
ticularly to the study of the form of movement of vibrating
FBWTON TO EULER. 243
strings, first reduced to mechanical principles by Mm. This
work contains also "Taylor s theorem/ the importance of
which was not recognised by analysts for over fifty years,
until Lagrange pointed out its power. His proof of it does not
consider the question of convergency, and is quite worthless.
The first rigorous proof was given a century later by Cauchy.
Taylor s work contains the first correct explanation of astro
nomical refraction. He wrote also a work on linear per
spective, a treatise which, like his other writings, suffers for
want of fulness and clearness of expression. At the age of
twenty-three he gave a remarkable solution of the problem of
the centre of oscillation, published in 1714. His claim to
priority was unjustly disputed by John Bernoulli.
Colin Maclaurin (1698-1746) was elected professor of mathe
matics at Aberdeen at the age of nineteen by competitive
examination, and in 1725 succeeded James Gregory at the Uni
versity of Edinburgh. He enjoyed the friendship of Newton,
and, inspired by Newton s discoveries, he published in 1719 his
Geometria Organica, containing a new and remarkable mode
of generating conies, known by his name. A second tract,
De Linearum geometricarum Proprietatibus, 1720, is remarkable
for the elegance of its demonstrations. It is based upon two
theorems : the first is the theorem of Cotes ; the second is
Maclaurin s : If through any point a line be drawn meeting
the curve in n points, and at these points tangents be drawn,
and if any other line through cut the curve in JS 13 R% etc.,
and the system of n tangents in r 1? r 2 , etc., then S -=S
OM OT
This and Cotes theorem are generalisations of theorems of
Newton. Maclaurin uses these in his treatment of curves of
the second and third degree, culminating in the remarkable
theorem that if a quadrangle has its vertices and the two
points of intersection of its opposite sides upon a curve of the
244 A HISTORY OF MATHEMATICS.
third degree, then, the tangents drawn at two opposite vertices
cut each other on the curve. He deduced independently
Pascal s theorem on the hexagram. The following is his ex
tension of this theorem (Phil Trans., 1735) : If a polygon
move so that each of its sides passes through a fixed point,
and if all its summits except one describe curves of the degrees
m, n, p, etc., respectively, then the free summit moves on a
curve of the degree 2 mnp , which reduces to mnp when
the fixed points all lie on a straight line. Maclaurin wrote on
pedal curves; He is the author of an Algebra. The object of
his treatise on Fluxions was to found the doctrine of fluxions
on geometric demonstrations after the manner of the ancients,
and thus, by rigorous exposition, answer such attacks as Berke
ley s that the doctrine rested on fals$ reasoning. The Fluxions
contained for the first time the correct way of distinguishing
between maxima and minima, and explained their use in the
theory of multiple points. "Maclaurin s theorem" was pre
viously given by James Stirling, and is but a particular case
of " Taylor s theorem." Appended to the treatise on Fluxions
is the solution of a number of beautiful geometric, mechanical,
and astronomical problems, in which he employs ancient
methods with such consummate skill as to induce Clairaut to
abandon analytic methods and to attack the problem of the
figure of the earth by pure geometry. His solutions com
manded the liveliest admiration of Lagrange. Maclaurin in
vestigated the attraction of the ellipsoid of revolution, and
showed that a homogeneous liquid mass revolving uniformly
around an axis under the action of gravity must assume the
form of an ellipsoid of revolution. Newton had given this
theorem without proof. Not withstanding the genius of Mac
laurin, his influence on the progress of mathematics in Great
Britain was unfortunate; for, by his example, he induced his
countrymen to neglect analysis and to be indifferent to the
NEWTON TO ETJLER. 245
wonderful progress in tlie Mglier analysis made on tlie Con
tinent.
It remains for us to speak of Abraham de Moivre (1667-1754),
who was of Erench descent, but was compelled to leave France
at the age of eighteen, on the Revocation of the Edict of Kantes.
He settled in London, where he gave lessons in mathematics.
He lived to the advanced age of eighty-seven and sank into a
state of almost total lethargy. His subsistence was latterly
dependent on the solution of questions on games of chance and
problems on probabilities, which he was in the habit of giving
at a tavern in St. Martin s Lane. Shortly before his death he
declared that it was necessary for him to sleep ten or twenty
minutes longer every day. The day after he had reached the
total of over twenty-three hours, he slept exactly twenty-four
hours and then passed away in his sleep. De Moivre enjoyed
the friendship of ISTewton and Halley. His power as a math
ematician lay in analytic rather than geometric investigation.
He revolutionised higher trigonometry by the discovery of the
theorem known by his name and by extending the theorems on
the multiplication and division of sectors from the circle to the
hyperbola. His work on the theory of probability surpasses
anything done by any other mathematician except Laplace.
His principal contributions are his investigations respecting
the Duration of Play, his Theory of Recurring Series, and
his extension of the value of Bernoulli s theorem by the aid
of Stirling s theorem. 42 His chief works are the Doctrine of
Chances, 1716, the Miscellanea Analytica, 1730, and Ms papers
in the Philosophical Transactions.
246 A HISTORY OF MATHEMATICS.
EULEB, LAGKANGE, AND LAPLACE.
During the epoch of ninety years from 1730 to 1820 the French
and Swiss cultivated mathematics with most brilliant success.
]STo previous period had shown such an array of illustrious
names. At this time Switzerland had her Euler ; France, her
Lagrange, Laplace, Legendre, and Monge. The mediocrity of
French mathematics which marked the time of Louis XIV.
was now followed by one of the. very brightest periods of all
history. England and Germany, on the other hand, which
during the unproductive period in France had their JSTewton
and Leibniz, could now boast of no great mathematician.
France now waved the mathematical sceptre. Mathematical
studies among the English and German people had sunk to
the lowest ebb. Among them the direction of original research
was ill-chosen. The former adhered with excessive partiality
to ancient geometrical methods ; the latter produced the com
binatorial school, which brought forth nothing of value.
The labours of Euler, Lagrange, and Laplace lay in higher
analysis, and this they developed to a wonderful degree. By
them analysis came to be completely severed from geometry.
During the preceding period the effort of mathematicians
not only in England, but, to some extent, even on the conti
nent, had been directed toward the solution of problems
clothed in geometric garb, and the results of calculation
were usually reduced to geometric form. A change now
took place. Euler brought about an emancipation of the
analytical calculus from geometry and established it as an
independent science. Lagrange and Laplace scrupulously
adhered to this separation. Building on the broad foun
dation laid for higher analysis and mechanics by jSTewton
and Leibniz, Euler, with matchless fertility of mind, erected
EULER, LAGRA3TG-E, AND LAPLACE. 247
an elaborate structure. There are few great ideas pursued
by succeeding analysts which were not suggested by Euler,
or of which he did not share the honour of invention.
With, perhaps, less exuberance of invention, but with more
comprehensive genius and profounder reasoning, Lagrange
developed the infinitesimal calculus and placed analytical
mechanics into the form in which we now know it. La
place applied the calculus and mechanics to the elaboration
of the theory of universal gravitation, and thus, largely ex
tending and supplementing the labours of ISTewton, gave a full
analytical discussion of the solar system. He also wrote an
epoch-marking work on Probability. Among the analytical
branches created during this period are the calculus of Varia
tions by Euler and Lagrange, Spherical Harmonics by La
place and Legendre, and Elliptic Integrals by Legendre.
Comparing the growth of analysis at this time with the
growth during the time of Gauss, Cauchy, and recent mathe
maticians, we observe an important difference. During the
former period we witness mainly a development with refer
ence to form. Placing almost implicit confidence in results of
calculation, mathematicians did not always pause to discover
rigorous proofs, and were thus led to general propositions,
some of which have since been found to be true in only special
cases. The Combinatorial School in Germany carried this
tendency to the greatest extreme ; they worshipped formalism
and paid no attention to the actual contents of formulae. But
in recent times there has been added to the dexterity in the
formal treatment of problems, a much-needed rigour of demon
stration. A good example of this increased rigour is seen in
the present use of infinite series as compared to that of Euler,
and of Lagrange in his earlier works.
The ostracism of geometry, brought about by the master
minds of this period, could not last permanently. Indeed, a
248 A HISTOBY OF MATHEMATICS.
new geometric school sprang into existence in France before
the close of this period. Lagrange would not permit a single
diagram to appear in his M&canique analytique, but thirteen
years before his death, Monge published his epoch-making
Gr&ometrie descriptive.
Leonhard Euler (1707-1783) was bom in Basel. His father,
a minister, gave him his first instruction in mathematics and
then sent him to the University of Basel, where he became a
favourite pupil of John Bernoulli. In his nineteenth year he
composed a dissertation on the masting of ships, which re
ceived the second prize from the French Academy of Sciences.
When John Bernoulli s two sons, Daniel and Nicolaus, went to
Russia, they induced Catharine I., in 1727, to invite their friend
Euler to St. Petersburg, where Daniel, in 1733, was assigned to
the chair of mathematics. In 1735 the solving of an astrono
mical problem, proposed by the Academy, for which several
eminent mathematicians had demanded some months time,
was achieved in three days by Euler with aid of improved
methods of his own. But the effort threw him into a fever
and deprived him of the use of his right eye. With still
superior methods this same problem was solved later by the
illustrious Gauss in one hour ! 47 The despotism of Anne I.
caused the gentle Euler to shrink from public affairs and to
devote all his time to science. After his call to Berlin by
Frederick the Great in 1747, the queen of Prussia, who
received him kindly, wondered how so distinguished a scholar
should be so timid and reticent. Euler naively replied,
" Madam, it is because I come from a country where, when one
speaks, one is hanged." In 1766 he with difficulty obtained
permission to depart from Berlin to accept a call by Catha
rine II. to St. Petersburg. Soon after his return to Eussia he
became blind, but this did not stop his wonderful literary
productiveness, which continued for seventeen years, until the
ETJLER, LAGRANGE, AND LAPLACE. 249
day of Ms death. 45 He dictated to Ms servant Ms Anleitung
zur Algebra, 1770, which, though purely elementary, is meri
torious as one of the earliest attempts to put the fundamental
processes on a sound basis.
Euler wrote an immense number of works, chief of which
are the following : Introductio in analysin injtnitorum. 1748,
a work that caused a revolution in analytical mathematics, a
subject which had hitherto never been presented in so general
and systematic manner ; Institutiones calculi differentialis, 1755,
and Institutiones calculi integraliSj 1768-1770, which were the
most complete and accurate works on the calculus of that time,
and contained not only a full summary of everything then
known on this subject, but also the Beta and Gamma Func
tions and other original investigations ; Methodus inveniendi
lineas curvas maximi minimive proprietate gaudentes, 1744,
which, displaying an amount of mathematical genius seldom
rivalled, contained his researches on the calculus of variations
(a subject afterwards improved by Lagrange), to the invention
of which Euler was led by the study of isoperimetrical curves,
the brachistochrone in a resisting medium, and the theory of
geodesies (subjects which had previously engaged the attention
* of the elder Bernoullis and others) ; the Theoria motuum plane-
tarum et cometarum, 1744, TJieoria motus lunce, 1753, TJieoria
motuum lunfje, 1772, are his chief works on astronomy ; Ses
lettres ct une princesse d Allemagne sur quelques sujets de
Physique et de Philosophic, 1770, was a work which enjoyed
great popularity.
We proceed to mention the principal innovations and inven
tions of Euler. He treated trigonometry as a branch of
analysis, introduced (simultaneously with Thomas Simpson in
England) the now current abbreviations for trigonometric
functions, and simplified formulae by the simple expedient
of designating the angles of a triangle by A, B, C, and the
250 A HISTORY OF MATHEMATICS.
opposite sides by a, &, c, respectively. He pointed out tlie
relation between trigonometric and exponential functions. In
a paper of 1737 we first meet the symbol IT to denote 3. 14159 -. 21
Euler laid down the rules for the transformation of co-ordinates
in space, gave a methodic analytic treatment of plane curves
and of surfaces of the second order. He was the first to
discuss the equation of the second degree in three variables,
and to classify the surfaces represented by it. By criteria
analogous to those used in the classification of conies he
obtained five species. He devised a method of- solving bi
quadratic equations by assuming x = Vp + V# + Vr, with the
hope that it would lead him to a general solution of algebraic
equations. The method of elimination by solving a series of
linear equations (invented independently by Bezout) and the
method of elimination by symmetric functions, are due to him. 20
Far reaching are Euler s researches on logarithms. Leibniz
and John Bernoulli once argued the question whether a
negative number has a logarithm. Bernoulli claimed that
since ( a) 2 = (+&) 2 ; we have log( a) 2 = log(+a) 2 and
21og( a) = 2 log(+ a), and finally log ( a) = log (+ a).
Euler proved that a has really an infinite number of loga
rithms, all of which are imaginary when a is negative, and all
except one when a is positive. He then explained how
log (a) 2 might equal log(+a) 2 , and yet log (a) not
equal log (+a).
The subject of infinite series received new life from Mm.
To his researches on series we owe the creation of the theory of
definite integrals by the development of the so-called Eulerian
integrals. He warns his readers occasionally against the use
of divergent series, but is nevertheless very careless himself.
The rigid treatment to which infinite series are subjected now
was then undreamed of. No clear notions existed as to what
constitutes a convergent series. Neither Leibniz nor Jacob
EULER, LAGEAISTGE, AND LAPLACE. 251
and John Bernoulli had entertained any serious doubt of the
correctness of the expression | = 1 1 + 1 !-{-.... Guido
Grandi went so far as to conclude from this that ^ = -}- -f-
+ . In the treatment of series Leibniz advanced a meta
physical method of proof which held sway over the minds of
the elder Bernoullis, and even of Euler. 46 The tendency of
that reasoning was to justify results which seem to us now
highly absurd. The looseness of treatment can best be seen
from examples. The very paper in which Euler cautions
against divergent series contains the proof that
+ - + 1 4- n + n 2 + ... = as follows :
n 2 n
1 % n n 2 n1
these added give zero. Euler has no hesitation to write
1 3 + 5 7 -\ ---- =0, and no one objected to such results
excepting Nicolaus Bernoulli, the nephew of John and Jacob.
Strange to say, Euler finally succeeded in converting Mcolaus
Bernoulli to his own erroneous views. At the present time
it is difficult to believe that Euler should have confidently
written sin <f> 2 sin 2 < + 3 sin 3 <j> 4 sin 4 <j> -) ---- = 0, but
such examples afford striking illustrations of the want- 1 of
scientific basis of certain parts of analysis at that time.
Euler s proof of the binomial formula for negative and
fractional exponents, which has been reproduced in elemen
tary text-books of even recent years, is faulty. A remarkable
development, due to Euler, is what he named the hypergeo-
metric series, the summation of which he observed to be
dependent upon the integration of a linear differential equa
tion of the second order, but it remained for Gauss to point
out that for special values of its letters, this series represented
nearly all functions then, known.
Euler developed the calculus of finite differences in the first
252 A HISTORY OF MATHEMATICS.
chapters of his Institutiones calculi differentialis, and then
deduced the differential calculus from it. He established a
theorem on homogeneous functions, known by his name, and
contributed largely to the theory of differential equations, a
subject which had received the attention of Newton, Leibniz,
and the Bernoullis, but was still undeveloped. Clairaut,
Fontaine, and Euler about the same time observed criteria of
integrability, but Euler in addition showed how to employ
them to determine integrating factors. The principles on
which the criteria rested involved some degree of obscurity.
The celebrated addition-theorem for elliptic integrals was first
established by Euler. He invented a new algorithm for
continued fractions, which he employed in the solution of
the indeterminate equation ace 4- by = c. We now know that
substantially the same solution of this equation was given
1000 years earlier, by the Hindoos. By giving the factors of
the number 2 2 " 1 + 1 when n = 5, he pointed out that this ex
pression did not always represent primes, as was supposed by
Fermat. He first supplied the proof to "Fermat s theorem,"
and to a second theorem of Fermat, which states that every
prime of the form 4n + l is expressible as the sum of two
squares in one and only one way. A third theorem of Fermat,
that x n + y n = z n , has no integral solution for values of n
greater than 2, was proved by Euler to be correct when n = 3.
Euler discovered four theorems which taken together make
out the great law of quadratic reciprocity, a law independently
discovered by Legendre. 48 Euler enunciated and proved a
well-known theorem, giving the relation between the number
of vertices, faces, and edges of certain polyhedra, which,
however, appears to have been known to Descartes. The
powers of Euler were directed also towards the fascinating
subject of the theory of probability, in which he solved some
difficult problems.
EULER, LAGRANGE, AKB LAPLACE. 253
Of no little importance are Eider s labours in analytical
mechanics. Says Whewell: "The person who did most to
give to analysis the generality and symmetry which are now
its pride, was also the person who made mechanics analytical ;
I mean Euler." n He worked out the theory of the rotation of
a body around a fixed point, established the general equations
of motion of a free body, and the general equation of hydrody
namics. He solved an immense number and variety of mechan
ical problems, which arose in his mind on all occasions. Thus,
on reading Virgil s lines, "The anchor drops, the rushing keel
is staid/ 5 he could not help inquiring what would be the
ship s motion in such a case. About the same time as paniel
Bernoulli he published the Principle of the Conservation of
Areas and defended the principle of "least action," advanced
by Maupertius. He wrote also on tides and on sound.
Astronomy owes to Euler the method of the variation of
arbitrary constants. By it he attacked the problem of per
turbations, explaining, in case of two planets, the secular vari
ations of eccentricities, nodes, etc. He was one of the first
to take up with success the theory of the moon s motion by
giving approximate solutions to the " problem of three bodies."
He laid a sound basis for the calculation of tables of the moon.
These researches on the moon s motion, which captured two
prizes, were carried on while he was blind, with the assistance
of his sons and two of his pupils.
Most of his memoirs are contained in the transactions of
the Academy of Sciences at St. Petersburg, and in those of
the Academy at Berlin. Erom 1728 to 1783 a large portion
of the Petropolitan transactions were filled by his writings.
He had engaged to furnish the Petersburg Academy with
memoirs in. sufficient number to enrich its acts for twenty
years a promise more than fulfilled, for down to 1818 the
volumes usually contained one or more papers of his. It has
254 A HISTORY OF MATHEMATICS.
been said that an edition of Euler 3 s complete works "would fill
16,000 quarto pages. His mode of working was, first to con
centrate Ms powers upon a special problem, then to solve
separately all problems growing out of the first. ISTo one
excelled him in dexterity of accommodating methods to special
problems. It is easy to see that mathematicians could not
long continue in Euler s habit of writing and publishing. The
material would soon grow to such enormous proportions as to
be unmanageable. We are not surprised to see almost the
opposite in Lagrange, his great successor. The great French
man delighted in the general and abstract, rather than, like
Euler, in the special and concrete. His writings are con
densed and give in a nutshell what Euler narrates at great
length.
Jean-le-Rond D Alembert (1717-1783) was exposed, when
an infant, by his mother in a market by the church of St.
Jean-le-Rond, near the Notre-Dame in Paris, from which he
derived his Christian name. He was brought up by the wife
of a poor glazier. It is said that when he began to show signs
of great talent, his mother sent for him, but received the
reply, "You are only my step-mother; the glazier s wife is
my mother." His father provided him with a yearly income.
D Alembert entered upon the study of law, but such was his
love for mathematics, that law was soon abandoned. At the
age of twenty-four his reputation as a mathematician secured
for him admission to the Academy of Sciences. In 1743
appeared his Traitt de dynamique, founded upon the important
general principle bearing his name : The impressed forces are
equivalent to the effective forces. D Alembert s principle
seems to have been recognised before him by Fontaine, and
in some measure by John Bernoulli and !N"ewton. D Alembert
gave it a clear mathematical form and made numerous appli
cations of it. It enabled the laws of motion and the reason-
ETJLBB, LAGRANGE, AND LAPLACE. 255
ings depending on them to be represented in the most general
form, in analytical language. D Alembert applied it in 1744
in a treatise on the equilibrium and motion of fluids, in 1746
to a treatise on the general causes of winds, which obtained
a prize from the Berlin Academy. In both these treatises, as
also in one of 1747, discussing the famous problem of vibrating
chords, he was led to partial differential equations. He was
a leader among the pioneers in the study of such equations.
To the equation ^f = a2 ^3 arising in the problem of vibrat
ing chords, he gave as the general solution,
and showed that there is only one arbitrary function, if y be
supposed to vanish for x = and x = I. Daniel Bernoulli,
starting with a particular integral given by Brook Taylor,
showed that this differential equation is satisfied by the
trigonometric series
.cos- + 0sin -cos + --^
i If V
and claimed this expression to be the most general solution.
Euler denied its generality, on the ground that, if true, the
doubtful conclusion would follow that the above series repre
sents any arbitrary function of a variable. These doubts were
dispelled by Fourier. Lagrange proceeded to find the sum
of the above series, but D Alembert rightly objected to his
process, on the ground that it involved divergent series. 46
A most beautiful result reached by D Alembert, with aid
of his principle, was the complete solution of the problem of
the precession of the equinoxes, which had baffled the talents
of the best minds. He sent to the French Academy in 1747,
on the same day with Clairaut, a solution of the problem of
three bodies. This had become a question of universal inter-
256 A HISTORY OJF MATHEMATICS.
est to mathematicians, in which, each, vied to outdo all others.
The problem of two bodies, requiring the determination of
their motion when they attract each other with forces in
versely proportional to the square of the distance between
them, had been completely solved by Newton. The " problem
of three bodies " asks for the motion of three bodies attracting
each other according to the law of gravitation. Thus far,
the complete solution of this has transcended the power of
analysis. The general differential equations of motion were
stated by Laplace, but the dimculty arises in their integration.
The "solutions" hitherto given are merely convenient methods
of approximation in special cases when one body is the sun,
disturbing the motion of- the moon around the earth, or where
a planet moves under the influence of the sun and another
planet.
In the discussion of the meaning of negative quantities, of
the fundamental processes of the calculus, and of the theory of
probability, D Alembert paid some attention to the philosophy
of mathematics. His criticisms were not always happy. In
1754 he was made permanent secretary of the French Academy.
During the last years of his life he was mainly occupied with
the great French encyclopaedia, which was begun by Diderot
and himself. D Alembert declined, in 1762, an invitation of
Catharine II. to undertake the education of her son. Frederick
the Great pressed him to go to Berlin. He made a visit, but
declined a permanent residence there.
Alexis Claude Clairaut (1713-1765) was a youthful prodigy.
He read PHospitaPs works on the infinitesimal calculus and on
conic sections at the age of ten. In 1731 was published his
Hecherches sur les courbes & double courbure, which he had ready
for the press when he was sixteen. It was a work of remark
able elegance and secured his admission to the Academy of
Sciences when still under legal age. In 1731 he gave a proof of
EULEB, LAGRANGE, AND LAPLACE. 257
the theorem enunciated by Newton, that every cubic is a pro
jection of one of five divergent parabolas. Glairaut formed the
acquaintance of Maupertius, whom he accompanied on an expe
dition to Lapland to measure the length of a degree of the
meridian. At that time the shape of the earth was a subject
of serious disagreement. Newton and Huygens had concluded
from theory that the earth was flattened at the poles. About
1713 Dominico Cassini measured an arc extending from Dunkirk
to Perpignan and arrived at the startling result that the earth
is elongated at the poles. To decide between the conflicting
opinions, measurements were renewed. Maupertius earned by
his work in Lapland the title of " earth flattener " by disprov
ing the Cassinian tenet that the earth was elongated at the
poles, and showing that Newton was right. On his return, in
1743, Clairaut published a work, TMorie de la figure de la Terre,
which was based on the results of Maclaurin on homogeneous
ellipsoids. It contains a remarkable theorem, named after
Clairaut, that the sum of the fractions expressing the ellipticity
and the increase of gravity at the pole is equal to 2J- times the
fraction expressing the centrifugal force at the equator, the
unit of force being represented by the force of gravity at the
equator. This theorem is independent of any hypothesis with
respect to the law of densities of the successive strata of the
earth. It embodies most of Clairaut s researches. Todhunter
says that "in the figure of the earth no other person has
accomplished so much as Clairaut, and the subject remains at
present substantially as he left it, though the form is different.
The splendid analysis which Laplace supplied, adorned but did
not really alter the theory which started from the creative
hands of Clairaut."
In 1752 he gained a prize of the St. Petersburg Academy
for his paper on Thforie de la Lune, in which for the first time
modern analysis is applied to lunar motion. This contained
258 A HISTOEY OF MATHEMATICS.
the explanation of the motion of the lunar apsides. This
motion, left unexplained by Newton, seemed to Mm at first
inexplicable by Newton s law, and he was on the point of
advancing a new hypothesis regarding gravitation, when, tak
ing the precaution to carry his calculation to a higher degree
of approximation, he reached results agreeing with observa
tion. The motion of the moon was studied about the same
time by Euler and D Alembert. Clairaut predicted that
"Halley s Comet," then expected to return, would arrive at
its nearest point to the sun on April 13, 1759, a date which
turned out to be one month too late. He was the first to
detect singular solutions in differential equations of the first
order but of higher degree than the first.
In their scientific labours there was between Clairaut and
D Alembert great rivalry, often far from friendly. The grow
ing ambition of Clairaut to shine in society, where he was a
great favourite, hindered his scientific work in the latter part
of his life.
Johann Heinrich Lambert (1728-1777), born at Muhlhausen
in Alsace, was the son of a poor tailor. While working at his
father s trade, he acquired through his own unaided efforts a
knowledge of elementary mathematics. At the age of thirty he
became tutor iii a Swiss family and secured leisure to continue
his studies. In his travels with his pupils through Europe he
became acquainted with the leading mathematicians. In 1764
he settled in Berlin, where he became member of the Academy,
and enjoyed the society of Euler and Lagrange. He received
a small pension, and later became editor of the Berlin JSphem-
em. His many-sided scholarship reminds one of Leibniz.
In his Oosmological Letters he made some remarkable prophe
cies regarding the stellar system. In mathematics he made
several discoveries which were extended and overshadowed by
his great contemporaries. His first research on pure mathe-
EULER, LAGRANGE, AND LAPLACE. 259
matics developed in an infinite series the root x of the equation
# j^px = q. Since each equation of the form aaf + boo 8 = d
can be reduced to x m + px = g in two ways, one or the other of
the two resulting series was always found to be convergent,
and to give a value of x. Lambert s results stimulated Euler,
who extended the method to an equation of four terms, and
particularly Lagrange, who found that a function of a root of
a _ x + <f> (x) = can be expressed by the series bearing his
name. In 1761 Lambert communicated to the Berlin Academy
a memoir, in which he proves that TT is irrational. This proof
is given in Note IV. of Legendre s Gfeometrie, where it is
extended to ?r 2 . To the genius of Lambert we owe the intro
duction into trigonometry of hyperbolic functions, which he
designated by sinli x, cosh x, etc. His Freye Perspective, 1759
and 1773, contains researches on descriptive geometry, and
entitle him to the honour of being the forerunner of Monge.
In his effort to simplify the calculation of cometary orbits, he
was led geometrically to some remarkable theorems on conies,
for instance this : " If in two ellipses having a common major
axis we take two such arcs that their chords are equal, and
that also the sums of the radii vectores, drawn respectively
from the foci to the extremities of these arcs, are equal to
each other, then the sectors formed in each ellipse by the arc
and the two radii vectores are to each other as the square
roots of the parameters of the ellipses." B
John Landen (1719-1790) was an English mathematician
whose writings served as the starting-point of investigations
by Euler, Lagrange, and Legendre. Landen s capital discov
ery, contained in a memoir of 1755, was that every arc of the
hyperbola is immediately rectified by means of two arcs of an
ellipse. In his "residual analysis" he attempted to obviate
the metaphysical difficulties of fluxions by adopting a purely
algebraic method. Lagrange s Oalcul des Fonctions is based
260 A HISTOBY OF MATHEMATICS.
upon this idea. Landen showed how the algebraic expression
for the roots of a cubic equation could be derived by applica
tion of the differential and integral calculus. Most of the
time of this suggestive writer was spent in the pursuits of
active life.
Etienne Bezout (1730-1783.) was a French writer of popular
mathematical school-books. In his TMorie gn6rale des Equa
tions Algflbriqu&s, 1779, he gave the method of elimination by
linear equations (invented also by Euler) . This method was
first published by him in a memoir of 1764, in which he uses
determinants, without, however, entering upon their theory.
A beautiful theorem as to the degree of the resultant goes by
his name.
Louis Arbogaste ^1759-1803) of Alsace was professor of
mathematics at Strasburg. His chief work, the Calcul des
Derivations^ 1800^ gives the method known by his name, by
which the successive coefficients of a development are derived
from one another when, the expression is complicated. De
Morgan has pointed out that the true nature of derivation
is differentiation accompanied by integration. In this book
for the first time are the symbols of operation separated from
those of quantity. The notation D x y for ~ is due to him.
Maria Gaetana Agnesi (1718-1799) of Milan, distinguished as
a linguist, mathematician, and philosopher, filled the mathe
matical chair at the University of Bologna during her father s
Sickness. In 1748 she published her Instituzioni Analiticlie>
which was translated into English in 1801. The "witch of
Agnesi " or " versiera " is a plane curve containing a straight
line, cc = 0, and a cubic f ^ ) +1=-.
Joseph Louis Lagrange (1736-1813), one of the greatesl
mathematicians of all times, was born at Turin and died a1
Paris. He was of French extraction. His father, who hac
EULEK, LAGRANGE, AND LAPLACE. 261
charge of the Sardinian military chest, was once wealthy, but
lost all he had in speculation. Lagrange considered this loss
his good fortune, for otherwise he might not have made math
ematics the pursuit of his life. While at the college in Turin
his genius did not at once talfe its true bent. Cicero and Vir
gil at first attracted him more than Archimedes and Newton.
He soon came to admire the geometry of the ancients, but the
perusal of a tract of Halley roused his enthusiasm for the
analytical method, in the development of which he was des
tined to reap undying glory. He now applied himself to
mathematics, and in his seventeenth year he became professor
of mathematics in the royal military academy at Turin.
Without assistance or guidance he entered upon a course of
study which in two years placed him on a level with the
greatest of his contemporaries. With aid of his pupils he
established a society which subsequently developed into the
Turin Academy. In the first five volumes of its transactions
appear most of his earlier papers. At the age of nineteen he
communicated to Euler a general method of dealing with
" isoperimetrical problems," known now* as the Calculus of
Variations. This commanded Euler s lively admiration, and
he courteously withheld for a time from publication some
researches of his own on this subject, so that the youthful
Lagrange might complete his investigations and claim the
invention. Lagrange did quite as much as Euler towards the
creation of the Calculus of Variations. As it came from Euler
it lacked an analytic foundation, and this Lagrange supplied.
He separated the principles of this calculus from geometric
considerations by which his predecessor had derived them.
Euler had assumed as fixed the limits of the integral, i.e. the
extremities of the curve to be determined, but Lagrange
removed this restriction and allowed all co-ordinates of the
curve to vary at tlie same time. Euler introduced in 1766 the
262 A HISTORY OF MATHEMATICS.
name " calculus of variations/ and did much, to improve this
science along the lines marked out by Lagrange.
Another subject engaging the attention of Lagrange at
Turin was the propagation of sound. In his papers on this
subject in the Miscellanea Taurinensia, the young mathemati
cian appears as the critic of Newton, and the arbiter between
Euler and D Alembert. By considering only the particles
which are in a straight line, he reduced the problem to the
same partial differential equation that represents the motions
of vibrating strings. The general integral of this was found by
D Alembert to contain two arbitrary functions, and the ques
tion now came to be discussed whether an arbitrary function
may be discontinuous. D Alembert maintained the negative
against Euler, Daniel Bernoulli, and finally Lagrange, argu
ing that in order to determine the position of a point of the
chord at a time t, the initial position of the chord must be
continuous. Lagrange settled the question in the affirmative.
By constant application during nine years, Lagrange, at the
age of twenty-six, stood at the summit of European fame.
But his intense studies had seriously weakened a constitution
never robust, and though his physicians induced him to take
rest and exercise, his nervous system never fully recovered its
tone, and he was thenceforth subject to fits of melancholy.
In 1764 the Trench Academy proposed as the subject of
a prize the theory of the libration of the moon. It demanded
an explanation, on the principle of universal gravitation, why
the moon always turns, with but slight variations, the same
phase to the earth. Lagrange secured the prize. This suc
cess encouraged the Academy to propose as a prize the theory
of the four satellites of Jupiter, a problem of six bodies,
more difficult than the one of three bodies previously solved
by Clairaut, D Alembert, and Euler. Lagrange overcame the
difficulties, but the shortness of time did, not permit him to
EULEE, LAGEANGE, AND LAPLACE. 263
exhaust the subject. Twenty-four years afterwards it was
completed by Laplace. Later astronomical investigations of
Lagrange are on coinetary perturbations (1778 and 1783), on
Kepler s problem, and on a new method of solving the prob
lem of three Bodies.
Being anxious to make the personal acquaintance of leading
mathematicians, Lagrange visited Paris, where he enjoyed the
stimulating delight of conversing with Clairaut, D Aleinbert,
Condor cet, the Abbe Marie, and others. He had planned a
visit to London, but he fell dangerously ill after a dinner in
Paris, and was compelled to return to Turin. In 1766 Euler
left Berlin for St. Petersburg, and he pointed out Lagrange as
the only man capable of filling the place. D Alembert recom
mended him at the same time. Frederick the Great there
upon sent a message to Turin, expressing the wish of "the
greatest king of Europe " to have " the greatest mathemati
cian " at his court. Lagrange went to Berlin, and staid there
twenty years. - Finding all his colleagues married, and being
assured by their wives that the marital state alone is happy,
he married. The union was not a happy one. His wife
soon died. Frederick the Great held him in high esteem,
and frequently conversed with him on the advantages of per
fect regularity of life. This led Lagrange to cultivate regular
habits. He worked no longer each day than experience taught
him he could without breaking down. His papers were care
fully thought out before he began writing, and wfren he wrote
he did so without a single correction.
During the twenty years in Berlin he crowded the transac
tions of the Berlin Academy with memoirs, and wrote also
the epoch-making work called the M6canique Analytique. He
enriched algebra by researches on the solution of equations.
There are two methods of solving directly algebraic equa
tions, that of substitution and that of combination. The
264 A HISTORY OF MATHEMATICS.
former method was developed by Ferrari, Vieta, Tchirnhausen,
Euler, Bezout, and Lagrange ; the latter by Vandermonde and
Lagrange. 20 In the method of substitution the original forms
are so transformed that the determination of the roots is made
to depend upon simpler functions (resolvents). In the method
of combination auxiliary quantities are substituted for certain
simple combinations ("types") of the unknown roots of the
equation, and auxiliary equations (resolvents) are obtained for
these quantities with aid of the coefficients of the given equa
tion. Lagrange traced all known algebraic solutions of equa
tions to the uniform principle consisting in the formation and
solution of equations of lower degree whose roots are linear
functions of the required roots, and of the roots of unity. He
showed that the quintic cannot be reduced in this way, its
resolvent being of the sixth degree. His researches on the
theory of equations were continued after he left Berlin. In
the Resolution des equations num6riques (1798) he gave a
method of approximating to the real roots of numerical equa
tions by continued fractions. Among other things, it contains
also a proof that every equation must have a root, a theorem
which appears before this to have been considered self-evident.
Other proofs of this were given by Argand, Gauss, and Cauchy.
In a note to the above work Lagrange uses Fermat s theorem
and certain suggestions of Gauss in effecting a complete alge
braic solution of any binomial equation.
While in Berlin Lagrange published several papers on the
theory of numbers. In 1769 he gave a solution in integers of
indeterminate equations of the second degree, which resembles
the Hindoo cyclic method ; he was the first to prove, in 1771,
"Wilson s theorem," enunciated by an Englishman, John
Wilson, and first published by Waring in his Meditationes
Algebraicce ; he investigated in 1775 under what conditions
2 and 5 (1 and 3 having been discussed by Euler)
EULER, LAGRANGE, AND LAPLACE. 265
are quadratic residues, or non-residues of odd prime numbers,
q ; he proved in 1770 Meziriac s theorem that every integer is
equal to the sum of four, or a less number, of squares. He
proved Ferinat s theorem on x n + y n = z n , for the case n = 4,
also Fermat s theorem that, if a 2 -j- 5 2 = e 2 , then ab is not a
square.
In his memoir on Pyramids, 1773, Lagrange made consider
able use of determinants of the third order, and demonstrated
that the square of a determinant is itself a determinant. He
never, however, dealt explicitly and directly with determi
nants; he simply obtained accidentally identities which are
now recognised as relations between determinants.
Lagrange wrote much on differential equations. Though
the subject of contemplation by the greatest mathematicians
(Euler, D Alembert, Clairaut, Lagrange, Laplace), yet more
than other branches of mathematics did they resist the sys
tematic application of fixed methods and principles. Lagrange
established criteria for singular solutions (Calcul des Fonctions,
Lessons 14-17), which are, however, erroneous. He was the
first to point out the geometrical significance of such solutions.
He generalised Euler s researches on total differential equa
tions of two variables, and of the ninth order ; he gave a solu
tion of partial differential equations of the first order (Berlin
Memoirs, 1772 and 1774), and spoke of their singular solutions,
extending their solution in Memoirs of 1779 and 1785 to equa
tions of any number of variables. The discussion on partial
differential equations of the second order, carried on by
D Alembert, Euler, and Lagrange, has already been referred
to in our account of D Alembert.
While in Berlin, Lagrange wrote the t M&cJianiqueAncdytique"
the greatest of his works (Paris, 1788). From the principle
of virtual velocities he deduced, with aid of the calculus of
variations, the whole system of mechanics so elegantly and
266 A HISTORY OF MATHEMATICS.
harmoniously that it may fitly be called, in Sir William
Kowan Hamilton s words, " a kind of scientific poem." It is a
most consummate example of analytic generality. Geometrical
figures are nowhere allowed. " On ne trouvera point de figures
dans cet ouvrage" (Preface). The two divisions of mechanics
statics and dynamics are in the first four sections of each
carried out analogously, and each is prefaced by a historic
sketch of principles. Lagrange formulated the principle of
least action. In their original form, the equations of motion
involve the co-ordinates x, y, z, of the different particles m or
dm of the system. But x, y, z, are in general not independent,
and Lagrange introduced in place of them any variables , fa
<, whatever, determining the position of the point at the time.
These may be taken to be independent. The equations of
motion may now assume the form
ddT dT, A.
or when H, \l/, <,... are the partial differential coefficients
with respect to , ^, <, . . . of one and the same function V,
then the form
__ , A
dt dg d d
The latter is par excellence the Lagrangian form of the equa
tions of motion. With Lagrange originated the remark that
mechanics may be regarded as a geometry of four dimensions.
To him falls the honour of the introduction of the potential
into dynamics. 49 Lagrange was anxious to have his Mfoanique
Analytique published in Paris. The work was ready for print,
in 1786, but not till 1788 could he find a publisher, and then
only with the condition that after a few years he would pur
chase all the unsold copies. The work was edited by
Legendre.
EULEB, LAGBANGE, AND LAPLACE. 267
After the death of Frederick the Great, men of science
were no longer respected in Germany, and Lagrange accepted
an invitation of Louis XVI. to migrate to Paris. The French
queen treated him with regard, and lodging was procured for
him in the Louvre. But he was seized with a long attack of
melancholy which destroyed his taste for mathematics. For
two years his printed copy of the Mtcanique, fresh from the
press., the work of a quarter of a century, lay unopened on
his desk. Through Lavoisier he became interested in chem
istry, which he found "as easy as algebra." The disastrous
crisis of the French Kevolution aroused him again to activity.
About this time the young and accomplished daughter of the
astronomer Lemonnier took compassion on the sad, lonely
Lagrange, and insisted upon marrying him. Her devotion to
him constituted the one tie to life which at the approach of
death he found it hard to break.
He was made one of the commissioners to establish weights
and measures having units founded on nature. Lagrange
strongly favoured the decimal subdivision, the general idea of
which was obtained from a work of Thomas Williams, London,
1788. Such was the moderation of Lagrange s character, and
such the universal respect for him, that he was retained as presi
dent of the commission on. weights and measures even after it
had been purified by the Jacobins by striking out the names
of Lavoisier, Laplace, and others. Lagrange took alarm at the
fate of Lavoisier, and planned to return to Berlin, but at the
establishment of the Ecole Normale in 1795 in Paris, he was
induced to accept a professorship. Scarcely had he time to
elucidate the foundations of arithmetic and algebra to young
pupils, when the school was closed. His additions to the
algebra of Euler were prepared at this time. In 1797 the
Ecole Polytechnique was founded, with Lagrange as one of
the professors. The earliest triumph of this institution was
268 A HISTORY OF MATHEMATICS.
the restoration of Lagrange to analysis. His mathematical
activity burst out anew. He brought forth the Theorie des
fonctions analytiques (1797), Legons sur le calcul des fonctions,
a treatise on the same lines as the preceding (1801), and the
Resolution des equations numeriques (1798). In 1810 he
began a thorough revision of his Mecanique analytique, but
he died before its completion.
The TJieorie des fonctions, the germ of which is found in a
memoir of his of 1772, aimed to place the principles of the
calculus upon a sound foundation by relieving the mind of the
difficult conception of a limit or infinitesimal. John Landen s
residual calculus, professing a similar object, was unknown to
him. Lagrange attempted to prove Taylor s theorem (the
power of which he was the first to point out) by simple algebra,
and then to develop the entire calculus from that theorem.
The principles of the calculus were in his day involved in
philosophic difficulties of a serious nature. The infinitesimals
of Leibniz had no satisfactory metaphysical basis. In the
differential calculus of Euler they were treated as absolute
zeros. In Newton s limiting ratio, the magnitudes of which it
is the ratio cannot be found, for at the moment when they
should be caught and equated, there is neither arc nor chord.
The chord and arc were not taken by Newton as equal before
vanishing, nor after vanishing, but when they vanish. " That
method," said Lagrange, "has the great inconvenience of con
sidering quantities in the state in which they cease, so to
speak, to be quantities ; for though we can always well con
ceive the ratios of two quantities, as long as they remain
finite, that ratio offers to the mind no clear and precise idea,
as soon as its terms become both nothing at the same time."
D Alembert s method of limits was much the same as the
method of prime and ultimate ratios. D Alembert taught
that a variable actually reached its limit. When Lagrange
EULEB, LAGRANGE, AND LAPLACE. 269
endeavoured to free the calculus of its metaphysical difficulties,
by resorting to common algebra, he avoided the whirlpool of
Charybdis only to suffer wreck against the rocks of Scylla.
The algebra of his day, as handed down to him by Euler, was
founded on a false view of infinity. ISTo correct theory of
infinite series had then been established. Lagrange proposed
to define the differential coefficient of /(a?) with respect to x
as the coefficient of h in the expansion of f(x + Ji) by Taylor s
theorem, and thus to avoid all reference to limits. But he
used infinite series without ascertaining that they were con
vergent, and his proof that f(x + h) can always be expanded
in a series of ascending powers of h, labours under serious
defects. Though Lagrange s method of developing the calculus
was at first greatly applauded, its defects were fatal, and to-day
his "method of derivatives," as it was called, has been gen
erally abandoned. He introduced a notation of his own, but
it was inconvenient, and was abandoned by him in the second
edition of his Mecanique, in which he used infinitesimals. The
primary object of the Theorie des fonctions was not attained,
but its secondary results were far-reaching. It was a purely
abstract mode of regarding functions, apart from geometrical
or mechanical considerations. In the further development
of higher analysis a function became the leading idea, and
Lagrange s work may be regarded as the starting-point of the
theory of functions as developed by Cauchy, Eiemann, Weier-
strass, and others.
In the treatment of infinite series Lagrange displayed in
his earlier writings that laxity common to all mathematicians
of his time, excepting Mcolaus Bernoulli II. and D Alembert.
But his later articles mark the beginning of a period of greater
rigour. Thus, in the Gakul de fonctions he gives his theorem
on the limits of Taylor s theorem. Lagrange s mathematical
researches extended to subjects which have not been men-
270 A HISTOKY OF MATHEMATICS.
tioned liere such as probabilities, finite differences, ascend
ing continued fractions, elliptic integrals. Everywhere his
wonderful powers of generalisation and abstraction are made
manifest. In that respect he stood without a peer, but
his great contemporary, Laplace, surpassed him in practical
sagacity. Lagrange was content to leave the application of
his general results to others, and some of the most important
researches of Laplace (particularly those on the velocity of
sound and on the secular acceleration of the moon) are im
plicitly contained in Lagrange s works.
Lagrange was an extremely modest man, eager to avoid
controversy, and even timid in conversation. He spoke in
tones of doubt, and his first words generally were, "Je ne
sais pas." He would never allow his portrait to be taken,
and the only ones that were secured were sketched without
his knowledge by persons attending the meetings of the
Institute.
Pierre Simon Laplace (1749-1827) was born at Beaumont-
en-Auge in Normandy. Yery little is known of his early
life. When at the height of his fame he was loath to speak
of his boyhood, spent in poverty. His father was a small
farmer. Some rich neighbours who recognised the boy s
talent assisted him in securing an -education. As an extern
he attended the military school in Beaumont, where at an
early age he became teacher of mathematics. At eighteen
he went to Paris, armed with letters of recommendation to
D Alembert, who was then at the height of his fame. The
letters remained -unnoticed, but young Laplace, undaunted,
wrote the great geometer a letter on the principles of me
chanics, which brought the following enthusiastic response:
" You needed no introduction ; you have recommended your
self ; my support is your due." D Alembert secured him a
position at the Ecole Militaire of Paris as professor of mathe-
EULEB, LAGRANGE, AND LAPLACE. 271
matics. His future was now assured, and he entered upon
those profound researches which brought him the title of "the
Newton of France." With wonderful mastery of analysis,
Laplace attacked the pending problems in the application
of the law of gravitation to celestial motions. During the
succeeding fifteen years appeared most of his original contri
butions to astronomy. His career was one of almost uninter
rupted prosperity. In 1784 he succeeded Bezout as examiner
to the royal artillery, and the following year he became mem
ber of the Academy of Sciences. He was made president of
the Bureau of Longitude j he aided in the introduction of the
decimal system, and taught, with Lagrange, mathematics in
the Ecole Normale. When, during the Bevolution, there arose
a cry for the reform of everything, even of the calendar,
Laplace suggested the adoption of an era beginning with the
year 1250, when, according to his calculation, the major axis
of the earth s orbit had been perpendicular to the equinoctial
line. The year was to begin with the vernal equinox, and the
zero meridian was to be located east of Paris by 185.30 degrees
of the centesimal division of the quadrant, for by this meridian
the beginning of his proposed era fell at midnight. But the
revolutionists rejected this scheme, and made the start of the
new era coincide with the beginning of the glorious French
Eepublic. 50
Laplace was justly admired throughout Europe as a most
sagacious and profound scientist, but, unhappily for his repu
tation, he strove not only after greatness in science, but also
after political honours. The political career of this eminent
scientist was stained by servility and suppleness. After the
18th of Brumaire, the day when Napoleon was made emperor,
Laplace s ardour for republican principles suddenly gave way
to a great devotion to the emperor. Napoleon rewarded this
devotion by giving him the post of minister of the interior,
272 A HISTORY OF MATHEMATICS.
but dismissed Mm after six months for incapacity. Said
Napoleon, " Laplace ne saisissait .aucune question sous son
veritable point de vue ; il cherchait des subtilites partout,
n avait que des idees problematiques, et portait enfin Pesprit
des infiniinent petits jusgue dans P administration." Desirous
to retain Ms allegiance, Napoleon elevated Mm to the Senate
and bestowed various other honours upon him. Nevertheless,
he cheerfully gave his voice in 1814 to the dethronement of
his patron and hastened to tender his services to the Bourbons,
thereby earning the title of marquis. This pettiness of his
character is seen in his writings. The first edition of the
Syst&me du monde was dedicated to the Council of Mve Hun
dred. To the third volume of the M6canique Celeste is prefixed
a note that of all the truths contained in the book, that most
precious to the author was the declaration he thus made of
gratitude and devotion to the peace-maker of Europe. After
this outburst of affection, we are surprised to find in the editions
of the Theorie analytique des probability which appeared after
the Restoration, that the original dedication to the emperor is
suppressed.
Though supple and servile in politics, it must be said that
in religion and science Laplace never misrepresented or con
cealed his own convictions however distasteful they might be
to others. In mathematics and astronomy his genius shines
with a lustre excelled by few. Three great works did he give
to the scientific world, the Mecanique Celeste, the Exposition
du systeme du monde, and the Theorie anatytique des probabili
ties. Besides these he contributed important memoirs to the
Prench Academy.
We first pass in brief review his astronomical researches.
In 1773 he brought out a paper in which he proved that the
mean motions or mean distances of planets are invariable or
merely subject to small periodic changes. This was the first
EULEJEt, LAGRANGE, AHD LAPLACE. 273
and most important step in establishing the stability of the
solar system. 51 To Newton and also to Euler it had seemed
doubtful whether forces so numerous, so variable in position,
so different in intensity, as those in the solar system, could be
capable of maintaining permanently a condition of equilibrium.
Newton was of the opinion that a powerful hand must inter
vene from time to time to repair the derangements occa
sioned by the mutual action of the different bodies. This
paper was the beginning of a series of profound researches by
Lagrange and Laplace on the limits of variation of the various
elements of planetary orbits, in which the two great mathema
ticians alternately surpassed and supplemented each other.
Laplace s first paper really grew out of researches on the
theory of Jupiter and Saturn. The behaviour of these planets
had been studied by Euler "and Lagrange without receiving
satisfactory explanation. Observation revealed the existence
of a steady acceleration of the mean motions of our moon and
of Jupiter and an equally strange diminution of the mean mo
tion of Saturn. It looked as though Saturn might eventually
leave the planetary system, while Jupiter would fall into the
sun, and the moon upon the earth. Laplace finally succeeded
in showing, in a paper of 1784-1786, that these variations
(called the "great inequality") belonged to the class of ordi
nary periodic perturbations, depending upon the law of attrac
tion. The cause of so influential a perturbation was found in
the commensurability of the mean motion of the two planets.
In the study of the Jovian system, Laplace was enabled to
determine the masses of the moons. He also discovered cer
tain very remarkable, simple relations between the movements
of those bodies, known as " Laws of Laplace." His theory of
these bodies was completed in papers of 1788 and 1789.
These, as well as the other papers here mentioned, were pub
lished in the Memoirs prfaentis par divers savans. The year
274 A HISTOKY OF MATHEMATICS.
1787 was made memorable by Laplace s announcement that
the lunar acceleration depended upon the secular changes in
the eccentricity of the earth s orbit. This removed all doubt
then existing as to the stability of the solar system. The uni
versal validity of the law of gravitation to explain all motion
in the solar system was established. That system, a then
known, was at last found to be a complete machine.
In 1796 Laplace published his Exposition du syst&me du
monde, a non-mathematical popular treatise on astronomy,
ending with a sketch of the history of the science. In this
work he enunciates for the first time his celebrated nebular
hypothesis. A similar theory had been previously proposed
by Kant in 1755, and by Swedenborg ; but Laplace does not
appear to have been aware of this.
Laplace conceived the idea of writing a work which should
contain a complete analytical solution of the mechanical prob
lem presented by the solar system, without deriving from
observation any but indispensable data. The result was the
Mtcaniq ue C&leste, which is a systematic presentation embrac
ing all the discoveries of Newton, Clairaut, D Alembert, Euler,
Lagrange, and of Laplace himself, on celestial mechanics.
The first and second volumes of this work were published in
1799 ; the third appeared in 1802, the fourth in 1805. Of the
fifth volume, Books XI. and XII. were published in 1823;
Books XIII., XIV., XV. in 1824, and Book XVI. in 1825. The
first two volumes contain the general theory of the motions
and figure of celestial bodies. The third and fourth volumes
give special theories of celestial motions, treating particu
larly of motions of comets, of our moon, and of other satel
lites. The fifth volume opens with a brief history of celestial
mechanics, and then gives in appendices the results of the
author s later researches. The Mcanique C&leste was such a
master-piece, and so complete, that Laplace s successors have
ETJLEB, LAGBANGE, AND LAPLACE. 275
been able to add comparatively little. The general part of
the work was translated into German by Joh. Karl Burk-
hardt, and appeared in Berlin, 1800-1802. Nathaniel Bowditch
brought out an edition in English, with an extensive com
mentary, in Boston, 1829-1839. The M6canique C6leste is not
easy reading. The difficulties lie, as a rule, not so much in the
subject itself as in the want of verbal explanation. A compli
cated chain of reasoning receives often no explanation what
ever. Biot, who assisted Laplace in revising the work for
the press, tells that he once asked Laplace some explanation
of a passage in the book which had been written not long
before, and that Laplace spent an hour endeavouring to recover
the reasoning which had been carelessly suppressed with the
remark, "II est facile de voir." Notwithstanding the impor
tant researches in the work, which are due to Laplace himself,
it naturally contains a great deal that is drawn from his pred
ecessors. It is, in fact, the organised result of a century of
patient toil. But Laplace frequently neglects to properly
acknowledge the source from which he draws, and lets the
reader infer that theorems and formulae due to a predecessor
are really his own.
We are told that when Laplace presented Napoleon with a
copy of the Mcamque Ctteste, the latter made the remark,
"M. Laplace, they tell me you have written this large book on
the system of the universe, and have never even mentioned
its Creator." Laplace is said to have replied bluntly, "Je
n avais pas besoin de cette hypothese-la." This assertion,
taken literally, is impious, but may it not have been intended
to convey a meaning somewhat different from its literal one ?
Newton was not able to explain by his law of gravitation all
questions arising in the mechanics of the heavens. Thus,
being unable to show that the solar system was stable, and
suspecting in fact that it was unstable, Newton expressed the
276 A HISTOBY OF MATHEMATICS.
opinion that tlie special intervention, from time to time, of a
powerful hand was necessary to preserve order. 3STow Laplace
was able to prove by the law of gravitation that the solar
system is stable, and in that sense may be said to have felt
no necessity for reference to the Almighty.
We now proceed to researches which belong more properly
to pure mathematics. Of these the most conspicuous are on
the theory of probability. Laplace has done more towards
advancing this subject than any one other investigator. He
published a series of papers, the main results of which were
collected in his TMorie anatytique des probabiliUs, 1812. The
third edition (1820) consists of an introduction and two books.
The introduction was published separately under the title,
Essai philosopliique sur les probability and is an admirable
and masterly exposition without the aid of analytical formulee
of the principles and applications of the science. The first
book contains the theory of generating functions, which are
applied, in the second book, to the theory of probability.
Laplace gives in his work on probability his method of
approximation to the values of definite integrals. The solu
tion of linear differential equations was reduced by him to
definite integrals. One of the most important parts of the
work is the application of probability to the method of least
squares, which is shown to give the most probable as well as
the most convenient results.
The first printed statement of the principle of least squares
was made in 1806 by Legendre, without demonstration. Gauss
had used it still earlier, but did not publish it until 1809.
The first deduction of the law of probability of error that
appeared in print was given in 1808 by Eobert Adrain in the
Analyst, a journal published by himself in Philadelphia. 2
Proofs of this law have since been given by G-auss, Ivory,
Herschel, Hagen, and others; but all proofs contain some
EULER, LAGRANOE, AND LAPLACE. 277
point of difficulty. Laplace s proof is perhaps the most satis
factory.
Laplace s work on probability is very difficult reading, par
ticularly the part on the method of least squares. The
analytical processes are by no means clearly established or
free from error. "No one was more sure of giving the
result of analytical processes correctly, and no one ever took
so little care to point out the various small considerations on
which correctness depends" (Be Morgan).
Of Laplace s papers on the attraction of ellipsoids, the most
important is the one published in 1785, and to a great extent
reprinted in the third volume of the M6canique Celeste. It
gives an exhaustive treatment of the general problem of
attraction of any ellipsoid upon a particle situated outside
or upon its surface. Spherical harmonics, or the so-called
"Laplace s coefficients," constitute a powerful analytic engine
in the theory of attraction, in electricity, and magnetism.
The theory of spherical harmonics for two dimensions had
been previously given by Legendre. Laplace failed to make
due acknowledgment of this, and there existed, in con
sequence, between the two great men, "a feeling more
than coldness." The potential function, V, is much used by
Laplace, and is shown by him to satisfy the partial differential
equation 5i + -11- + _- = 0. This is known as Laplace s
dx? dy 2 dz*
equation, and was first given by him in the more complicated
form which it assumes in polar co-ordinates. The notion
of potential was, however, not introduced into analysis by
Laplace. The honour of that achievement belongs to La-
grange. 49
Among the minor discoveries of Laplace are his method of
solving equations of the second, third, and fourth degrees,
his memoir on singular solutions of differential equations, Ms
278 A HISTORY OF MATHEMATICS.
researches in finite differences and in determinants, the estab
lishment of the expansion theorem in determinants which had
been previously given by Vanderrnonde for a special case, the
determination of the complete integral of the linear differen
tial equation of the second order. In the Mecaniqite Celeste he
made a generalisation of Lagrange s theorem on the develop
ment of functions in series known as Laplace s theorem.
Laplace s investigations in physics were quite extensive.
We mention here his correction of Newton s formula on the
velocity of sound in gases by taking into account the changes
of elasticity due to the heat of compression and cold of rarefac
tion ; his researches on the theory of tides ; his mathematical
theory of capillarity ; his explanation of astronomical refrac
tion ; his formulae for measuring heights by the barometer.
Laplace s writings stand out in bold contrast to those of
Lagrange in their lack of elegance and symmetry. Laplace
looked upon mathematics as the tool for the solution of physi
cal problems. The true result being once reached, he spent
little time in explaining the various steps of his analysis, or
in polishing his work. The last years of his life were spent
mostly at Arcueil in peaceful retirement on a country-place,
where he pursued his studies with his usual vigour until his
death. He was a great admirer of Euler, and would often
say, "Lisez Euler, lisez Euler, c est notre maitre a tous,"
Abnit-TfceopMle Vandermonde (1735-1796) studied music
during his youth in Paris and advocated the theory that all
art rested upon one general law, through which any one could
become a composer with the aid of mathematics. t He was the
first to give a connected and logical exposition of the theory
of determinants, and may, therefore, almost be regarded as
the founder of that theory. He and Lagrange originated the
method of combinations in solving equations. 20 ,
Adrien Marie Legendre (1752-1833) was educated at the
EULEB, LAGRANGE, AND LAPLACE. 279
College Mazarin in Paris, wliere lie began the study of mathe
matics under Abbe Marie. His mathematical genius secured
for him the position of professor of mathematics at the mili
tary school of Paris. While there he prepared an essay on
the curve described by projectiles thrown into resisting media
(ballistic curve) , which captured a prize offered by the Eoyal
Academy -of Berlin. In 1780 he resigned his position in order
to reserve more time for the stiidy of higher mathematics.
He was then made member of several public commissions.
In 1795 he was elected professor at the E"ormal School and
later was appointed to some minor government" positions.
Owing to his timidity and to Laplace s unfriendliness toward
him, but few important public offices commensurate with his
ability were tendered to him.
As an analyst, second only to Laplace and Lagrange, Legen-
dre enriched mathematics by important contributions, mainly
on elliptic integrals, theory of numbers, attraction of ellip
soids, and least squares. The most important of Legendre s
works is his Fonctions elliptiques, issued in two volumes in
1825 and 1826. He took up the subject where Euler, Landen,
and Lagrange had left it, and for forty years was the only one
to cultivate this new branch of analysis, until at last Jacobi and
Abel stepped in with admirable new discoveries. 52 Legendre
imparted to the subject that connection and arrangement
which belongs to an independent science. Starting with an
integral depending upon the square root of a polynomial of
the fourth degree in x, he showed that such integrals can be
brought back to three canonical forms, designated by .F(<),
), and !!(<), the radical being expressed in the form
=Vi ^sin 2 ^. He also undertook the prodigious task
of calculating tables of arcs of the ellipse for different degrees
of amplitude and eccentricity, which supply the means of
integrating a large number of differentials.
280 A HISTORY OF MATHEMATICS.
An earlier publication which, contained part of his researches
on elliptic functions was his Oalcul integral in three volumes
(1811, 1816, 1817), in which he treats also at length of the
two classes of definite integrals named by him Eulerian. He
tabulated the values of log T(p) for values of p between
1 and 2.
One of the earliest subjects of research was the attraction
of spheroids, which suggested to Legendre the function P n)
named after him. His memoir was presented to the Academy
of Sciences in 1783. The researches of Maclaurin and Lagrange
suppose the point attracted by a spheroid to be at the surface
or within the spheroid, but Legendre showed that in order to
determine the attraction of a spheroid on any external point
it suffices to cause the surface of another spheroid described
upon the same foci to pass through that point. Other memoirs
on ellipsoids appeared later.
The two household gods to which Legendre sacrificed with
ever-renewed pleasure in the silence of his closet were the
elliptic functions and the theory of numbers. His researches
on the latter subject, together with the numerous scattered
fragments on the theory of numbers due to his predecessors
in this line, were arranged as far as possible into a systematic
whole, and published in two large quarto volumes, entitled
TIi6orie des nombres, 1830. Before the publication of this
work Legendre had issued at divers times preliminary articles.
Its crowning pinnacle is the theorem of quadratic reciprocity,
previously indistinctly given by Euler without proof, but for the
first time clearly enunciated and partly proved by Legendre. 48
While acting as one of the commissioners to connect Green
wich and Paris geodetically, Legendre calculated all the tri
angles in France. This furnished the occasion of establishing
formulae and theorems on geodesies, on the treatment of the
spherical triangle as if it were a plane triangle, by applying
EULER, LAGRANGE, AND LAPLACE. 281
certain corrections to the angles, and on the method of least
squares, published for the first time by him without demon
stration in 1806.
Legendre wrote an Elements de G-eometrie, 1794, which
enjoyed great popularity, being generally adopted on the
Continent and in the United States as a substitute for Euclid.
This great modern rival of Euclid passed through numerous
editions ; the later ones containing the elements of trigonom
etry and a proof of the irrationality of ir and -jr 2 . Much
attention was given by Legendre to the subject of parallel
lines. In the earlier editions of the Elements, he made direct
appeal to the senses for the correctness of the " parallel-axiom."
He then attempted to demonstrate that "axiom," but his
proofs did not satisfy even himself. In Vol. XII. of the
Memoirs of the Institute is a paper by Legendre, containing
his last attempt at a solution of the problem. Assuming
space to be infinite, he proved satisfactorily that it is impossible
for the sum of the three angles of a triangle to exceed two
right angles; and that if there be any triangle the sum of
whose angles is two right angles, then the same must be true
of all triangles. But in the next step, to show that this sum
cannot be less than two right angles, his demonstration neces
sarily failed. If it could be granted that the sum of the three
angles is always equal to two right angles, then the theory of
parallels could be strictly deduced.
Joseph Fourier (1768-1830) was born at Auxerre, in central
France. He became an orphan in his eighth year. Through
the influence of friends he was admitted into the military
school in his native place, then conducted by the Benedictines
of the Convent of St. Mark. He there prosecuted his studies,
particularly mathematics, with surprising success. He wished
to enter the artillery, but, being of low birth (the son of a
tailor), his application was answered thus: f Fourier, not
282 A HISTOBY OF MATHEMATICS.
being noble, could not enter the artillery, although he were
a second Newton." 53 He was soon appointed to the mathe
matical chair in the military school. At the age of twenty-
one he went to Paris to read before the Academy of Sciences
a memoir on the resolution of numerical equations, which
was an improvement on Newton s method of approximation.
This investigation of his early youth he never lost sight of.
He lectured upon it in the Polytechnic School ; he developed
it on the banks of the Nile ; it constituted a part of a work
entitled Analyse des equationes determines (1831), which was
in press when death overtook him. This* work contained
" Fourier s theorem" on the number of real roots between
two chosen limits. Budan had published this result as early
as 1807, but there is evidence to show that Fourier had estab
lished it before Sudan s publication. These brilliant results
were eclipsed by the theorem of Sturm, published in 1835.
Fourier took a prominent part at his home in promoting
the Eevolution. Under the French Eevolution the arts -and
sciences seemed for a time to flourish. The reformation of
the weights and measures was planned with grandeur of con
ception. The Normal School was created in 1795, of which
Fourier became at first pupil, then lecturer. His brilliant
success secured him a chair in the Polytechnic School, the
duties of which he afterwards quitted, along with Monge and
Berthollet, to accompany Napoleon on his campaign to Egypt.
Napoleon founded the Institute of Egypt, of which Fourier
became secretary. In Egypt he engaged not only in scientific
work, but discharged important political functions. After
his return to France he held for fourteen years the prefecture
of Grenoble. During this period he carried on his elaborate
investigations on the propagation of heat in solid bodies,
published in 1822 in his work entitled La Theorie Analytique
de la Ohaleur. This work marks an epoch in the history of
EULER, LAGRANGE, AND LAPLACE. 283
mathematical physics. "Fourier s series" constitutes its
gem. By this research a long controversy was brought to a
close, and the fact established that any arbitrary function
can be represented by a trigonometric series. The first
announcement of this great discovery was made by Fourier
in 1807, before the French Academy. The trigonometric
= eo
series S (a n sin nx + b n cos nx) represents the function <j> (#)
ft=0 I /*7T
for every value of x, if the coefficients a n =- \ <(V) sinnxdx,
7T*x JT
and & n be equal to a similar integral. The weak point in
Fourier s analysis lies in his failure to prove generally that
the trigonometric series actually converges to the value of
the function. In 1827 Fourier succeeded Laplace as president
of the council of the Polytechnic School.
Before proceeding to the origin of modern geometry we shall
speak briefly of the introduction of higher analysis into Great
Britain. This took place during the first quarter of this cen
tury. The British began to deplore the very small progress
that science was making in England as compared with its
racing progress on the Continent. In 1813 the "Analytical
Society" was formed at Cambridge. This was a small club
established by George Peacock, John Herschel, Charles Bab-
bage, and a few other Cambridge students, to promote, as it
was humorously expressed, the principles of pure "D-ism,"
that is, the Leibniziau notation in the calculus against those
of "dot-age," or of the Newtonian notation. This struggle
ended in the introduction into Cambridge of the notation
^, to the exclusion of the fluxional notation y. This
dx
was a great step in advance, not on account of any great
superiority of the Leibnizian over the Newtonian notation,
but because the adoption of the former opened up to English
students the vast storehouses of continental discoveries. Sir
William Thomson, Tait, and some other" modern writers find
284 A HISTORY OF MATHEMATICS.
it frequently convenient to use both, notations. Hersehel,
Peacock, and Babbage translated, in 1816, from the French,
Lacroix s treatise on the differential and integral calculus, and
added in 1820 two volumes of examples. Lacroix s was one
of the best and most extensive works on the calculus of that
time. Of the three founders of the "Analytical Society,"
Peacock afterwards did most work in pure mathematics.
Babbage became famous for his invention of a calculating
engine superior to Pascal s. It was never finished, owing
to a misunderstanding with the government, and a conse
quent failure to secure funds. John Herschel, the eminent
astronomer, displayed his mastery over higher analysis in
memoirs communicated to the Koyal Society on new applica
tions of mathematical analysis, and in articles contributed
to cyclopaedias on light, on meteorology, and on the history
of mathematics.
George Peacock (1791-1858) was educated at Trinity College,
Cambridge, became Lowndean professor there, and later, dean
of Ely. His chief publications are his Algebra, 1830 and 1842,
and his Report on Recent Progress in Analysis, which was the
first of several valuable summaries of scientific progress printed
in the volumes of the British Association. He was one of the
first to study seriously the fundamental principles of algebra,
and to fully recognise its purely symbolic character. He
advances, though somewhat imperfectly, the "principle of the
permanence of equivalent forms." It assumes that the rules
applying to the symbols of arithmetical algebra apply also
in symbolical algebra. About this time D. F. Gregory wrote
a paper "on the real nature of symbolical algebra," which
brought out clearly the commutative and distributive laws.
These laws had been noticed years before by the inventors
of symbolic methods in the calculus. It was Servois who
introduced the names commutative and distributive in 1813.
EULER, LAG-BANGE, AND LAPLACE. 285
Peacock s investigations on the foundation of algebra were
considerably advanced by De Morgan and Hankel.
James Ivory (1765-1845) was a Scotch, mathematician who
for twelve years, beginning in 1804, held the mathematical
chair in the Eoyal Military College at Marlow (now at Sand
hurst). He was essentially a self-trained mathematician, and
almost the only one in Great Britain previous to the organisa
tion of the Analytical Society who was well versed in conti
nental mathematics. Of importance is his memoir (Phil.
Trans., 1809) in which the problem of the attraction of a
homogeneous ellipsoid upon an external point is reduced to
the simpler problem of the attraction of a related ellipsoid
upon a corresponding point interior to it. This is known as
"Ivory s theorem." He criticised with undue severity Laplace s
solution of the method of least squares, and gave three proofs
of the principle without recourse to probability ; but they are
far from being satisfactory.
The Origin of Modern Geometry.
By the researches of Descartes and the invention of the cal
culus, the analytical treatment of geometry was brought into
great prominence for over a century. Notwithstanding the
efforts to revive synthetic methods made by Desargues, Pas
cal, De Lahire, ISTewton, and Maclaurin, the analytical method
retained almost undisputed supremacy. It was reserved for
the genius of Monge to bring synthetic geometry in the
foreground, and to open up new avenues of progress. His
Gom6trie descriptive marks the beginning of a wonderful
development of modern geometry.
Of the two leading problems of descriptive geometry, the
one to represent by drawings geometrical magnitudes was
brought to a high degree of perfection before the time of
286 A HISTORY OF MATHEMATICS.
Monge; tlie other to solve problems on figures in space
by constructions in a plane had received considerable at
tention before Ms time. His most noteworthy predecessor
in descriptive geometry was the Frenchman Frezier (1682-
1773). But it remained for Monge to create descriptive
geometry as a distinct branch of science by imparting to it
geometric generality and elegance. All problems previously
treated in a special and uncertain - manner were referred
back to a few general principles. He introduced the line
of intersection of the horizontal and the vertical plane as
the axis of projection. By revolving one plane into the
other around this axis or ground-line, many advantages were
gained. 54
G-aspard Monge (1746-1818) was born at Beaune. The con
struction of a plan of his native town brought the boy under
the notice of a colonel of engineers, who procured for him an
appointment in the college of engineers at Mezieres. Being
of low birth, he could not receive a commission in the army,
but he was permitted to enter the annex of the school, where
surveying and drawing were taught. Observing that all the
operations connected with the construction of plans of fortifi
cation were conducted by long arithmetical processes, he sub
stituted a geometrical method, which the commandant at first
refused even to look at, so short was the time in which it
could be practised ; when once examined, it was received with
avidity. Monge developed these methods further and thus
created his descriptive geometry. Owing to the rivalry
between the French military schools of that time, he was not
permitted to divulge his new methods to any one outside of
this institution. In 1768 he was made professor of mathemat
ics at Mezi&res. In 1780, when conversing with two of his
pupils, S. F. Lacroix and G-ayvernon in Paris, he was obliged
to say, " All that I have here done by calculation, I could have
EULER, LAGRANGE, AND LAPLACE. 287
done with, the ruler and compass, but I am not allowed to
reveal these secrets to you." But Lacroix set himself to
examine what the secret could be, discovered the processes, and
published them in 1795. The method was published by Monge
himself in the same year, first in the form in which the short
hand writers took down his lessons given at the Normal School,
where he had been elected professor, and then again, in revised
form, in the Journal des 6coles normdles. The next edition
occurred in 1798-1799. After an ephemeral existence of only
four months the Normal School was closed in 1795. In the
same year the Polytechnic School was opened, in the estab
lishing of which Monge took active part. He taught there
descriptive geometry until his departure from France to accom
pany Napoleon on the Egyptian campaign. He was the first
president of the Institute of Egypt. Monge was a zealous
partisan of Napoleon and was, for that reason, deprived of all
his honours by Louis XVIII. This and the destruction of the
Polytechnic School preyed heavily upon his mind. He did
not long survive this insult.
Mongers numerous papers were by no means confined to de
scriptive geometry. His analytical discoveries are hardly less
remarkable. He introduced into analytic geometry the me
thodic use of the equation of a line. He made important
contributions to surfaces of the second degree (previously
studied by Wren and Euler) and discovered between the
theory of surfaces and the integration of partial differential
equations, a hidden relation which threw new light upon both
subjects. He gave the differential of curves of curvature,
established a general theory of curvature, and applied it to the
ellipsoid. He found that the validity of solutions was not
impaired when imaginaries are involved among subsidiary
quantities. Mxmge published the following books: Statics,
1786 i Applications de I alg&bre a la g6om6trie, 1805 ; Applica-
288 A HISTORY OF MATHEMATICS.
tion de Vanalyse a la g6omtrie. The last two contain most of
his miscellaneous papers.
Monge was an inspiring teacher, and he gathered around
him a large circle of pupils, among which were Dupin, Servois,
Brianchion, Hachette, Biot, and Poncelet.
Charles Bupin (1784-1873), for many years professor of
mechanics in the Conservatoire des Arts et Metiers in Paris,
puhlished in 1813 an important work on Developpements de
gfometrie, in which is introduced the conception of conjugate
tangents of a point of a surface, and of the indicatrix. 55 It
contains also the theorem known as "Dupin s theorem."
Surfaces of the second degree and descriptive geometry were
successfully studied by Jean Nicolas Pierre Hachette (1769-
1834), who became professor of descriptive geometry at the
Polytechnic School after the departure of Monge for Rome and
Egypt. In 1822 he published his Traite de geometrie descriptive.
Descriptive geometry, which arose, as we have seen, in
technical schools in France, was transferred to Germany at
the foundation of technical schools there. G. Schreiber,
professor in Karlsruhe, was the first to spread Monge s
geometry in Germany by the publication of a work thereon
in 1828-1829. 54 In the United States descriptive geometry was
introduced in 1816 at the Military Academy in West Point
by Claude Crozet, once a pupil at the Polytechnic School in
Paris. Crozet wrote the first English work on the subject. 2
Lazare Nicholas Marguerite Carnot (1753-1823) was born at
ISTolay in Burgundy, and educated in his native province.
He entered the army, but continued his mathematical studies,
and wrote in 1784 a work on machines, containing the earliest
proof that kinetic energy is lost in collisions of bodies. With
the advent of the Eevolution he threw himself into politics,
and when coalesced Europe, in 1793, launched against France
a million soldiers, the gigantic task of organising fourteen
ETJLEB, LAGBANGE, AKD LAPLACE. 289
armies to meet the enemy was achieved by him. He was
banished in 1T96 for opposing Napoleon s coup d etat. The
refugee went to Geneva, where he issued, in 1797, a work
still frequently quoted, entitled, Reflexions sur la Metaphysique
du Oalcul Infinitesimal. He declared himself as an "irrecon
cilable enemy of kings." After the Eussian campaign he
offered to fight for France, though not for the empire. On
the restoration he was exiled. He died in Magdeburg. His
Geom6trie de position, 1803, and his Essay on Transversals,
1806, are important contributions to modern geometry. While
Monge revelled mainly in three-dimensional geometry, Carnot
confined himself to that of two. By his effort to explain
s the meaning of the negative sign in geometry he established
a " geometry of position," which, however, is different from
the "Geometric der Lage" of to-day. He invented a class
of general theorems on projective properties of figures, which
have since been pushed to great extent by Poncelet, Chasles,
and others.
Jean Victor Poncelet (1788-1867), a native of Metz, took
part in the Eussian campaign, was abandoned as dead on the
bloody field of Krasnoi, and taken prisoner to Saratoff. De
prived there of all books, and reduced to the remembrance
of what he had learned at the Lyceum at Metz and the Poly
technic School, where he had studied with predilection the
works of Monge, Carnot, and Brianchion, he began to study
mathematics from its elements. He entered upon original
researches which afterwards made him illustrious. While
in prison he did for mathematics what Bunyan did for
literature, produced a much-read work, which has remained
of great value down to the present time. He returned to
Prance in 1814, and in 1822 published the work in question,
entitled, Traiti des Proprietes projectives des figures. In it
he investigated the properties of figures which remain un-
290 A HISTOEY OF MATHEMATICS.
altered by projection of the figures. The projection is not
effected here by parallel rays of prescribed direction, as with
Monge, but by central projection. Thus perspective projec
tion; used before him by Desargues, Pascal, Newton, and Lam
bert, was elevated by him into a fruitful geometric method.
In the same way he elaborated some ideas of De Lahire,
Servois, and Gergonne into a regular method the method
of "reciprocal polars." To him we owe the Law of Duality
as a consequence of reciprocal polars. As an independent
principle it is due to G-ergonne. Poncelet wrote much on
applied mechanics. In 1838 the Faculty of Sciences was
enlarged by Ms election to the chair of mechanics.
While in France the school of Monge was creating modern
geometry, efforts were made in England to revive Greek
geometry by Robert Simson (1687-1768) and Matthew Stewart
(1717-1785). Stewart was a pupil of Simson and Maclaurin,
and succeeded the latter in the chair at Edinburgh. During
the eighteenth century he and Maclaurin were the only promi
nent mathematicians in Great Britain. His genius was ill-
directed by the fashion then prevalent in England to ignore
higher analysis. In his Four Tracts, Physical and Mathe
matical, 1761, he applied geometry to the solution of difficult
astronomical problems, which on the Continent were ap
proached analytically with greater success. He published, in
1746, General Theorems, and in 1763, his Propositions geo
metries more veterum demonstrator. The former work con
tains sixty-nine theorems, of which only five are accompanied
by demonstrations. It gives many interesting new results
on the circle and the straight line. Stewart extended some
theorems on transversals due to Giovanni Ceva (1648-1737),
an Italian, who published in 1678 at Mediolani a work con
taining the theorem now known by his name.
RECENT TIMES.
more zealously and successfully lias mathematics
been cultivated than in this century. Nor has progress, as
in previous periods, been confined to one or two countries.
"While the French and Swiss, who alone during the preceding
epoch carried the torch of progress, have continued to develop
mathematics with great success, from other countries whole
armies of enthusiastic workers have wheeled into the front
rank. Germany awoke from her lethargy by bringing for
ward G-auss, Jacobi, Dirichlet, and hosts of more recent men ;
Great Britain produced her De Morgan, Boolq, Hamilton,
besides champions who are still living; Eussia entered the
arena with her Lobatchewsky; Norway with Abel; Italy with
Cremona ; Hungary with her two Bolyais ; the United States
with Benjamin Peirce.
The productiveness of modern writers has been enormous.
" It is difficult," says Professor Cayley, 56 " to give an idea of
the vast extent of modern mathematics. This word extent J
is not the right one : I mean extent crowded witjj. beautiful
detail, not an extent of mere uniformity such as an object
less plain, but of a tract of beautiful country seen at first in
the distance, but which will bear to be rambled through and
studied in every detail of hillside and valley, stream, rock,
wood, and flower." It is pleasant to the mathematician to
think that in his, as in no other science, the achievements of
291
292 A HISTORY OF MATHEMATICS.
svery age remai-i possessions forever ; new discoveries seldom
disprove older tenets ; seldom is anything lost or wasted.
If it be asked wherein the utility of some modern exten
sions of mathematics lies, it must be acknowledged that it is
at present difficult to see how they are ever to become appli
cable to questions of common life or physical science. But
our inability to do this should not be urged as an argument
against the pursuit of such studies. In the first place, we
know neither the day nor the hour when these abstract
developments will find application in the mechanic arts, in
physical science, or in other branches of mathematics. For
example, the whole subject of graphical statics, so useful
to the practical engineer, was made to rest upon von Staudt s
Geometrie der Lags; Hamilton s " principle of varying action"
has its use in astronomy; complex quantities, general inte
grals, and general theorems in integration offer advantages in
the study of electricity and magnetism. " The utility of such
researches/ says Spottiswoode, 57 "can in no case be discounted,
or even imagined beforehand. Who, for instance, would have
supposed that the calculus of forms or the theory of substitu
tions would have thrown much light upon ordinary equations ;
or that Abelian functions and hyperelliptic transcendents
would have told us anything about the properties of curves ;
or that the calculus of operations would have helped us in
any way towards the figure of the earth ? " A second reason
in favour of the pursuit of advanced mathematics, even when
there is n^ promise of practical application, is this, that math
ematics, like poetry and music, deserves cultivation for its
own sake.
The great characteristic of modern mathematics is its gln-
eralising tendency. Nowadays little weight is given to iso
lated theorems, " except as affording hints of an unsuspected
new sphere of thought, like meteorites detached from some
SYNTHETIC GEOMETRY. 293
undiscovered planetary orb of speculation." In mathematics,
as in all true sciences, no subject is considered in itself alone,
but always as related to, or an outgrowth, of, other things.
The development of the notion of continuity plays a leading
part in modern research. In geometry the principle of con
tinuity, the idea of correspondence, and the theory of projec
tion constitute the fundamental modern notions. Continuity
asserts itself in a most striking way in relation to the circular
points at infinity in a plane. In algebra the modern idea finds
expression in the theory of linear transformations and invari
ants, and in the recognition of the value of homogeneity and
symmetry.
SYNTHETIC GEOMETRY.
The conflict between geometry and analysis which arose
near the close of the last century and the beginning of the
present has now come to an end. Neither side has come
out victorious. The greatest strength is found to lie, not in
the suppression of either, but in the friendly rivalry between
the two, and in the stimulating influence of the one upon the
other. Lagrange prided himself that in his Mecanique Ana-
lytique he had succeeded in avoiding all figures ; but since his
time mechanics has received much help from geometry.
Modern synthetic geometry was_created by several investi
gators about the same time. It seemed to be the outgrowth
of a desire for general^methods which should serve as threads
of Ariadne to guide the student through the labyrinth of theo
rems, corollaries, porisms, and problems. Synthetic geometry
wa^ first cultivated by Monge, Carnot, and Poncelet in France;
it then bore rich fruits at the hands of Mobius and Steiner in
Germany, and Switzerland, and was finally developed to still
294 A HISTOEY OF MATHEMATICS.
Mgher perfection by Chasles in France, yon Staudt in Ger
many, and Cremona in Italy.
Augustus Ferdinand Mobius (1790-1868) was a native of
Schulpforta in Prussia. He studied at Gottingen under
Gauss, also at Leipzig and Halle. In Leipzig he became, in
1815, privat-docent, the next year extraordinary professor of
astronomy, and in 1844 ordinary professor. This position he
held till his death. The most important of his researches
are on geometry. They appeared in Crelle s Journal, and in
his celebrated -work entitled Der BarycentriscJie Calcul, Leipzig,
1827. As the name indicates, this calculus is based upon
properties of the centre of gravity. 58 Thus, that the point
S is the centre of gravity of weights a, b, c, d placed at the
points A, B, C 9 D respectively, is expressed by the equation
(a + & + c + d) S = aA + IB + cO + dD.
His calculus is the beginning of a quadruple\lgebra, and con
tains the germs of Grassmann s marvellous system. In desig
nating segments of lines we find throughout this work for the
first time consistency in the distinction of positive and nega
tive by the order of letters AB, BA. Similarly for triangles
and tetrahedra. The remark that it is always possible to give
three points A 9 J3, such weights a, /3, y that any fourth point
M in their plane will become a centre of mass, led Mobius
to a new system of co-ordinates in which the position of a
point was indicated by an equation, and that of a line by
co-ordinates. By this algorithm he found by algebra many
geometric theorems expressing mainly invariantal properties,
for example, the theorems on the anharmonic relation.
Mobius wrote also on statics and astronomy. He generalised
spherical trigonometry by letting the sides or angles of tri
angles exceed 180.
SYNTHETIC GEOMETRY. 295
Jacob Steiner (1796-1863) , "the greatest geometrician since
the time of Euclid/ 3 was born in Utzendorf in the Canton of
Bern. He did not -learn to write till he was fourteen. At
eighteen he became a pupil of Pestalozzi. Later he studied
at Heidelberg and* Berlin. When Orelle started, in 1826, the
celebrated mathematical journal bearing his name, Steiner and
Abel became leading contributors. In 1832 Steiner published
his Systematische Entwickelung der AWiangigkeit geometrischer
Qestalten von einander, "in which is uncovered the organism
by which the most diverse phenomena (Erscheinungeri) in
the world of space are united to each other." Through the
influence of Jaeobi and others, the chair of geometry was
founded for him Li, Berlin in 1834. This position he occupied
until his death, which occurred after years of bad health. In
his Systematische Entwickelung en, for the first time, is the
principle of duality introduced at the outset. This book and
von Staudt s lay the foundation on which synthetic geometry
in its present form rests. Not only did he fairly complete the
theory of curves and surfaces of the second degree, but he
made great advances in the theory of those of higher degrees.
In his hands synthetic geometry made prodigious progress.
New discoveries followed each other so rapidly that he often
did not take time to record their demonstrations. In an
article in Orelle^s Journal on Allgemeine Eigenschaften Alge-
braischer Curven he gives without proof theorems which were
declared by Hesse to be "like Fermat s theorems, riddles to
the present and future generations." Analytical proofs of
some of them have been given since by others, but Cremona
finally proved them all by a synthetic method. Steiner dis
covered synthetically the two prominent properties of a sur
face of the third order; viz. that it contains twenty-seven
straight lines and a pentahedron which has the double points
for its vertices and the lines of tlie Hessian of the given sur-
296 A HISTORY OF MATHEMATICS.
face for its edges. 55 The first property was discovered ana
lytically somewhat earlier in England by Cayley and Salmon,
and -the second by Sylvester. Steiner s work on this subject
was the starting-point of important researches by H. Schroter,
If. August^ L. Cremona, and R. Sturm. Steiner made investi
gations by synthetic methods on maxima and minima, and
arrived at the solution of problems which at that time alto
gether surpassed the analytic power .of the calculus of varia
tions. He generalised the Jiexagrammum mysticum and also
MalfattPs problem. 59 Malfatti, in 1803, proposed the problem,
to cut three cylindrical holes out of a three-sided prism in
such a way that the cylinders and the prism have the same
altitude and that the volume of the cylinders be a maximum.
This problem was reduced to another, now generally known
as Malfatti 7 s problem: to inscribe three circles in a triangle
that each circle will be tangent to two sides of a triangle and
to the other two circles. Malfatti gave an analytical solution,-
but Steiner gave without proof a construction, remarked that
there were thirty-two solutions, generalised the problem by
replacing the three lines by three circles, and solved the
analogous problem for three dimensions. This general prob
lem was solved analytically by C. H. Schellbach (1809-1892)
and Cayley, and by Clebsch with the aid of the addition,
theorem of elliptic functions. 60
Steiner s researches are confined to synthetic geometry. He
hated analysis as thoroughly as Lagrange disliked geometry.
Steiner s Gesammelte WerTce were published in Berlin in 1881
and 1882.
Michel Chasles (1793-1880) was bom at Epernon, entered
the Polytechnic School of Paris in 1812, engaged afterwards
in business, which he later gave up that he might devote all
his time to scientific pursuits. In 1841 he became professor of
geodesy and mechanics at the Polytechnic School; later,
SYNTHETIC GEOMETRY. 297
Professeur de Geometric suprieure & la Faculte des Sciences
de Paris." He was a voluminous writer on geometrical sub
jects. In 1837 lie published Ms admirable Apergu historique
sur Vorigine et le developpement des metliodes en geometric, con
taining a history of geometry and, as an appendix, a treatise
"sur deux principes generaux de la Science." The Apergu
historique is still a standard historical work; the appendix
contains the general theory of Homography (Collineation) and
of duality (Reciprocity). The name duality is due to Joseph
Diaz Gergonne (1771-1859). Chasles introduced the term
axih^^ corresponding to the German Doppelver-
Mltniss and to Clifford s cross-ratio. Chasles and Steiner
elaborated independently the modern synthetic or projective
geometry. Numerous original memoirs of Chasles were pub
lished later in the Journal de VEcole Polytechnique. He gave
a reduction of cubics, different from Newton s in this, that the
five curves from which all others can be projected are sym
metrical with respect to a centre. In 1864 he began the pub
lication, in the Comptes rendus, of articles in which he solves
by his " method of characteristics " and the " principle of cor
respondence" an immense number of problems. He deter
mined, for instance, the number of intersections of two curves
in a plane. The method of characteristics contains the basis
of enumerative geometry. The application of the principle of
correspondence was extended by Cayley, A. Brill, H. G. Zeu-
then, H. A. Schwarz, G. H. Halphen (1844-1889), and others.
The full value of these principles of Chasles was not brought
out until the appearance, in 1879, of the Kalkul der Abzahl-
enden Geometrie by Hermann Schubert of Hamburg. This
work contains a masterly discussion of the problem of enumer
ative geometry, viz. to determine how many geometric figures
of given definition satisfy a sufficient number of conditions.
Schubert extended Ms enumerative geometry to n-dimensional
space. 55
298 A HISTOliY Olf MATHEMATICS.
To Chasles we owe the introduction into projective geometry
of non-pro jective properties of figures by means of the infi
nitely distant imaginary sphere-circle. 61 Eemarkable is his
complete solution, in 1846, by synthetic geometry, of the
difficult question of the attraction of an ellipsoid on an exter
nal point. This "was accomplished analytically by Poisson in
1835. The labours of Chasles and Steiner raised synthetic
geometry to an honoured and respected position by the side
of analysis.
Karl Georg Christian von Staudt (1798-1867) was born in
Eothenburg on the Tauber, and, at his death, was professor
in Erlangen. His great works are the Geometric der Lage,
Nurnberg, 1847, and his Beitrdge zur Geometric der Lage, 1856-
1860. The author cut loose from algebraic formulae and from
metrical relations, particularly the anharmonic ratio of Steiner
and Chasles, and then created a geometry of position, which
is a complete science in itself, independent of all measure
ments. He shows that projective properties of figures have
no dependence whatever on measurements, and can be estab
lished without any mention of them. In his theory of what
he calls "Wurfe," he even gives a geometrical definition of
a number in its relation to geometry as determining the posi
tion of a point. The Beitrdge contains the first complete and
general theory of imaginary points, lines, and planes in pro
jective geometry. Eepresentation of an imaginary point is
sought in the combination of an involution with a determi
nate direction, both on the real line through the point.
While purely projective, von Staudt s method is intimately
related to the problem of representing by actual points and
lines the imaginaries of analytical geometry. This was sys
tematically undertaken by C. IT. Maximilien Marie, who worked,
however, on entirely different lines. An independent attempt
lias been made recently (1893) by P. H. Loud of Colorado
SYNTHETIC GEOMETRY. 299
College. Yon Staudt s geometry of position was for a long
time disregarded; mainly, no doubt, because Ms book is
extremely condensed. An impulse to the study of this subject
was given by Culmann, who rests his graphical statics upon
the work of von Staudt. An interpreter of von Staudt was
at last found in Theodor Eeye of Strassburg, who wrote a
Geometric der Lage in 1868.
Synthetic geometry has been studied with much success by
Luigi Cremona, professor in the University of Borne. In
his Introduzione ad una teoria geometrica dette curve plane
he developed by a uniform method many new results and
proved synthetically all important results reached before that
time by analysis. His writings have been translated into
German by M. Curtze, professor at the gymnasium in Thorn.
The theory of the transformation of curves and of the corre
spondence of points on curves was extended by him to three
dimensions. Kuled surfaces, surfaces of the second order,
space-curves of the third order, and the general theory of
surfaces have received much attention at his hands.
Karl Culmann, professor at the Polytechnicum in Zurich,
published an epoch-making work on Die grapMsche Statik,
Zurich, 1864, which has rendered graphical statics a great
rival of analytical statics. Before Culmann, B. E. Gousinery
had turned his attention to the graphical calculus, but he
made use of perspective, and not of modern geometry. 62 Cul
mann is the first to undertake to present the graphical calculus
as a symmetrical whole, holding the same relation to the new
geometry that analytical mechanics does to higher analysis.
He makes use of the polar theory of reciprocal figures as
expressing the relation between the force and the funicular
polygons. He deduces this relation without leaving the plane
of the two figures. But if the polygons be regarded as pro
jections of lines in space, these lines may be treated as recipro-
300 A HISTOBY OF MATHEMATICS.
cal elements of a " Nullsystem." This was done by Clerk
Maxwell in 1S64, and elaborated further by Cremona.* The
graphical calculus has been applied by 0. MoTir of Dresden
to the elastic line for continuoiis spans. Henry T. Eddy, of
the Eose Polytechnic Institute, gives graphical solutions of
problems on the maximum stresses in bridges under concen
trated loads, with aid of what he calls "reaction polygons."
A standard work, La Statique grapliique, 1874, was issued by
Maurice Levy of Paris.
Descriptive geometry (reduced to a science by Monge in
France, and elaborated further by his successors, Hachette,
Dupin, Olivier, J. de la Gournerie) was soon studied also in
other countries. The French directed their attention mainly
to the theory of surfaces and their curvature ; the Germans
and Swiss, through Schreiber, Pohlke, Schlessinger, and par
ticularly Fiedler, interwove projective and descriptive geome
try. Bellavitis in Italy worked along the same line. The
theory of shades and shadows was first investigated by the
French writers just quoted, and in Germany treated most
exhaustively by Burmester. 62
During the present century very remarkable generalisations
have been made, which reach to the very root of two of the
oldest branches of mathematics, elementary algebra and
geometry. In algebra the laws of operation have been ex
tended; in geometry the axioms have been searched to the
bottom, and the conclusion has been reached that the space
defined by Euclid s axioms is not the only possible non-
contradictory space. Euclid proved (I. 27) that " if a straight
line falling on two other straight lines make the alternate
angles equal to one another, the two straight lines shall be
parallel to one another." Being unable to prove that in every
other case the two lines are not parallel, he assumed this to
be true in what is generally called the 12th " axiom," by some
SYNTHETIC GEOMETBY. 301
But this so-called axiom is far from
axiomatic. After centuries of desperate but fruitless attempts
to prove Euclid s assumption, the bold idea dawned upon
the minds of several mathematicians that a geometry might
be built up without assuming the parallel-axiom. While
Legendre still endeavoured to establish the axiom by rigid
proof, Lobatchewsky brought out a publication which assumed
the contradictory of that axiom, and which was the first of
a series of articles destined to clear up obscurities in the
fundamental concepts, and to greatly extend the field of
geometry.
Mcholaus Ivanovitch Lobatchewsky (1793-1856) was born at
Makarief, in Mschni-lSFowgorod, Kussia, studied at Kasan, and
from 1827 to 1846 was professor and rector of the University
of Kasan. His views on the foundation of geometry were
first made public in a discourse before the physical and mathe
matical faculty at Kasan, and first printed in the Kasan
Messenger for 1829, and then in the Gelelirte Schriften der
Universitdt Jasan, 1836-1838, under the title, " Tew Elements
of Geometry, with a complete theory of Parallels." Being
in the Baissian language, the work remained unknown to
foreigners, but even at home it attracted no notice. In 1840
he published a brief statement of his researches in Berlin.
Lobatchewsky constructed an "imaginary geometry," as he
called it, which has been described by Clifford as "quite
simple, merely Euclid without the vicious assumption." A
remarkable part of this geometry is this, that through a
point an indefinite number of lines can be drawn in a plane,
none of which cut a given line in the same plane. A similar
system of geometry was deduced independently by the Bolyais
in Hungary, who called it "absolute geometry."
Wolfgang Bolyai de Bolya (1775-1856) was born in Szekler-
Land, Transylvania. After studying at Jena, he went to
302 A HISTOBY OF MATHEMATICS.
Gottingen, where lie became intimate with. Gauss, then nine
teen years old. Gauss used to say that Bolyai was the only
man who fully understood his views on the metaphysics of
mathematics. Bolyai became professor at the Reformed Col
lege of Maros-V^sarhely, where for forty-seven years he had
for his pupils most of the present professors of Transylvania.
The first publications of this remarkable genius were dramas
and poetry. Clad in old-time planter s garb, he was truly
original in his private life as well as in his mode of thinking.
He was extremely modest. No monument, said he, should
stand over his grave, only an apple-tree, in memory of the
three apples ; the two of Eve and Paris, which made hell out
of earth, and that of Newton, which elevated the earth again
into the circle of heavenly bodies. 64 His son, Johann Bolyai
(1802-1860), was educated for the army, and distinguished
himself as a profound mathematician, an impassioned violin-
player, and an expert fencer. He once accepted the challenge
of thirteen officers on condition that after each duel he might
play a piece on his violin, and he vanquished them all.
The chief mathematical work of Wolfgang Bolyai appeared
in two volumes, 1832-1833, entitled Tentamen juventutem
studiosam in elementa matJieseos puree . . . introducendi. It
is followed by an appendix composed by his son Johann on
The Science Absolute of /Space. Its twenty-six pages make the
name of Johann Bolyai immortal. He published nothing else,
but he left behind one thousand pages of manuscript which
have never been read by a competent mathematician ! His
father seems to have been the only person in Hungary who
really appreciated the merits of his son s work. For thirty-
five years this appendix, as also Lobatchewsky s researches,
remained in almost entire oblivion. Finally Eichard Baltzer
of the University of Giessen, in 1867, called attention to the
wonderful researches. Johann Bolyai s Science Absolute of
SYNTHETIC GEOMETRY. 303
Space and Lobatchewsky s Geometrical Researches on the
Theory of Parallels (1840) were rendered easily accessible to
American readers by translations into English made in 1891
by George Bruce Halsted of the University of Texas.
The Kussian and Hungarian mathematicians were not the
only ones to whom pangeometry suggested itself. A copy of
the Tentamen reached Gauss, the elder Bolyai s former room
mate at Gottingen, and this Nestor of German mathematicians
was surprised to discover in it worked out what he himself
had begun long before, only to leave it after him in his
papers. As early as 1792 he had started on researches of that
character. His letters show that in 1799 he was trying to
prove a priori the reality of Euclid s system ; but some time
within the next thirty years he arrived at the conclusion
reached by Lobatchewsky and Bolyai. In 1829 he wrote to
Bessel, stating that his "conviction that we cannot found
geometry completely a priori has become, if possible, still
firmer," and that "if number is merely a product of our
mind, space has also a reality beyond our mind of which we
cannot fully foreordain the laws a priori." The term non-
Euclidean geometry is due to Gauss. It has recently been
brought to notice that Geronimo Saccheri, a Jesuit father of
Milan, in 1733 anticipated Lobatchewsky s doctrine of the
parallel angle. Moreover, G. B. Halsted has pointed out
that in 1766 Lambert wrote a paper " Zur Theorie der Parallel-
linien," published in the Leipziger Magazin fur reine und
angewandte MathematiJc, 1786, in which: (1) The failure of
the parallel-axiom in surface-spherics gives a geometry with
angle-sum > 2 right angles ; (2) In order to make intuitive
a geometry with angle-sum < 2 right angles we need the aid
of an "imaginary sphere" (pseudo-sphere); (3) In a space
with the angle-sum differing from 2 right angles, there is
an absolute measure (Bolyai s natural unit for length).
304 A HISTOBY OF MATHEMATICS.
In 1854, nearly twenty years later, G-auss heard from his
pupil, Riemann, a marvellous dissertation carrying the dis
cussion one step further by developing the notion of n-ply
extended magnitude, and the measure-relations of which a
manifoldness of n dimensions is capable, on the assumption
that every line may be measured by every other. Biernann
applied his ideas to space. He taught us to distinguish
between " unboundedness ;? and "infinite extent." According
to him we have in our mind a more general notion of space,
i.e. a notion of non-Euclidean space , but we learn by experience
that our physical space is, if not exactly, at least to high
degree of approximation, Euclidean space. Biemann s pro
found dissertation was not published until 1867, when it
appeared in the Gfottingen AbJiandlungen. Before this the
idea of n dimensions had suggested itself under various
aspects to Lagrange, Plucker, and H. Grassmann. About the
same time with Biemann s paper, others were published from
the pens of HelmJioltz and Beltrami. These contributed pow
erfully to the victory of logic over excessive empiricism. This
period marks the beginning of lively discussions upon this sub
ject. Some writers Bellavitis, for example were able to
see in non-Euclidean geometry and n-dimensional space noth
ing but huge caricatures, or diseased outgrowths of mathe
matics. Helmholtz s article was entitled Tliatsacken, welche
der Geometrie m Grunde liegen, 1868, and contained many of
the ideas of Biemann. Helmholtz popularised the subject in
lectures, and in articles for various magazines.
Eugenio Beltrami, born at Cremona, Italy, in 1835, and now
professor at Borne, wrote the classical paper Saggio di inter-
pretazione della geometria non-eudidea (Giorn. di Matem., 6),
which is analytical (and, like several other papers, should be
mentioned elsewhere "were we to adhere to a strict separation
between synthesis and analysis). He reached the brilliant
SYNTHETIC GBOMETEY. 305
and surprising conclnsion that the theorems of non-Euclidean
geometry rind their realisation upon surfaces of constant nega
tive curvature. He studied, also, surfaces of constant positive
curvature, and ended with the interesting theorem that the
space of constant positive curvature is contained in the space
of constant negative curvature. These researches of Beltrami,
Helinholtz, and Eiemann culminated in the conclusion that
on surfaces of constant curvature we may have three geome
tries, the non-Euclidean on a surface of constant negative
curvature, the spherical on a surface of constant positive cur
vature, and the Euclidean geometry on a surface of zero curva
ture. The three geometries do not contradict each other, but
are members of a system, a geometrical trinity. The ideas
of hyper-space were brilliantly expounded and popularised in
England by Clifford.
William Kingdon Clifford (1845-1879) was born at Exeter,
educated at Trinity College, Cambridge, and from 1871 until
his death professor of applied mathematics in University Col
lege, London. His premature death left incomplete several
brilliant researches which he had entered upon. Among these
are his paper On Classification of Loci and his Theory of
Graphs. He wrote articles On the Canonical Form and
Dissection of a Riemann s Surface, on Biguaternions, and
an incomplete work on the Elements of Dynamic. The
theory of polars of curves and surfaces was generalised by
him and by Eeye. His classification of loci, 1878, being a
general study of curves, was an introduction to the study
of n-dimensional space in a direction mainly projective.
This study has been continued since chiefly by G. Veronese
of Padua, C. Segre of Turin, E. Bertini, F. Aschieri, P. Del
Pezzo of Naples.
Beltrami ? s researches on non-Euclidean geometry were fol
lowed, in 1871, by important investigations of Felix Klein,
306 A HISTOBY OF MATHEMATICS.
resting upon Cayley s Sixth Memoir on Quantics, 1859. The
question whether it is not possible to so express the metrical
properties of figures that they will not vary by projection (or
linear transformation) had been solved for special projections
by Chasles, Poncelet, and E. Laguerre (1834-1886) of Paris,
but it remained for Cayley to give a general solution by denn
ing the distance between two points as an arbitrary constant
multiplied by the logarithm of the anharmonic ratio in which
the line joining the two points is divided by the fundamental
quadric. Enlarging upon this notion, Klein showed the inde
pendence of projective geometry from the parallel-axiom, and
by properly choosing the law of the measurement of distance
deduced from projective geometry the spherical, Euclidean,
and pseudospherical geometries, named by him respectively
the elliptic, parabolic, and hyperbolic geometries. This sug
gestive investigation was followed up by numerous writers,
particularly by G. Battaglini of Naples, E. d? Ovidio of Turin,
R. de Paolis of Pisa, F. Aschieri, A. Cayley, F. Lindemann
of Munich, E. Schering of Gottingen, W. Story of Clark
University, H. Stahl of Tubingen, A. Yoss of Wiirzburg,
Homersham Cox, A. Buchheim. 55 The geometry of n dimen
sions was studied along a line mainly metrical by a host of
writers, among whom may be mentioned Simon Newcomb of
the Johns Hopkins University, L. Schlafli of Bern, W. I.
Stringham of the University of California, W. Killing of
Minister, T. Craig of the Johns Hopkins, E. Lipschitz of
Bonn. E. S. Heath and Killing investigated the kinematics
and mechanics of such a space. Eegular solids in n-dimen-
sional space " were studied by Stringham, Ellery W. Davis
of the University of Nebraska, E. Hoppe of Berlin, and
others. Stringham gave pictures of projections upon our
space of regular solids in four dimensions, and Schlegel at
Hagen constructed models of such projections. These are
ANALYTIC GEOMETRY. 307
among the most curious of a series of models published by
L. Brill in Darmstadt. It has been pointed out that if a
fourth dimension existed, certain motions could take place
which we hold to be impossible. Thus INewcomb showed the
possibility of turning a closed material shell inside out by sim
ple flexure without either stretching or tearing ; Klein pointed
out that knots could not be tied 5 Veronese showed that a
body could be removed from a closed room without breaking
the walls ; C. S. Peirce proved that a body in four-fold space
either rotates about two axes at once, or cannot rotate without
losing one of its dimensions.
ANALYTIC GEOMETRY.
In the preceding chapter we endeavoured to give a flash
light view of the rapid advance of synthetic geometry. In
connection with hyperspace we also mentioned analytical
treatises. Modern synthetic and modern analytical geome
try have much in common, and may be grouped together
under the common name "protective geometry." Each has
advantages over the other. The continual direct viewing of
figures as existing in space adds exceptional charm to the
study of the former, but the latter has the advantage in this,
that a well-established routine in a certain degree may outrun
thought itself, and thereby aid original research. While in
Germany Steiner and von Staudt developed synthetic geome
try, Pliicker laid the foundation of modern analytic geometry.
Julius Pliicker (1801-1868) was born at Elberfeld, in Prus
sia. After studying at Bonn, Berlin, and Heidelberg, he spent
a short time in Paris attending lectures of Monge and his
pupils. Between 1826 and 1836 he held positions successively
at Bonn, Berlin, and Halle. He then became professor of
308 A HISTORY OF MATHEMATICS.
, physics at Bonn. Until 1846 his original researches were on
geometry. In 1828 and in 1831 he published his Anatytisch-
GeometriscJie Untersuchungen in two volumes. Therein he
adopted the abbreviated notation (used before Mm in a more
restricted way by Bobillier), and avoided the tedious process
of algebraic elimination by a geometric consideration. In the
second volume the principle of duality is formulated analyti
cally. With him duality and homogeneity found expression
already in his system of co-ordinates. The homogenous or
tri-linear system used by him is much the same as the co-or
dinates of Mobius. In the identity of analytical operation
and geometric construction Pliicker looked for the source of
Ms proofs. The System der Analytischen Geometrie, 1835, con
tains a complete classification of plane curves of the third
order, based on the nature of the points at infinity. The
Theorie der Algebraischen Curven, 1839, contains, besides an
enumeration of curves of the fourth order, the analytic rela
tions between the ordinary singularities of plane curves
known, as "Plucker s equations," by which he was able to
explain "Poncelet s paradox." The discovery of these rela
tions is, says Cayley, "the most important one beyond all
comparison in the entire subject of modern geometry." But
in Germany Plucker s researches met with no favour. His
method was declared to be unproductive as compared with
the synthetic method of Steiner and Poncelet! His rela
tions with Jacobi were not altogether friendly. Steiner once
declared that he would stop writing for Crelle s Journal if
Pliicker continued to contribute to it. 66 The result was that
many of Plucker s researches were published in foreign jour
nals, and that his work came to be better known in France
and England than in his native country. The charge was
also brought against Plucker that, though occupying the chair
of physics, he was no physicist. This induced him to relin-
ANALYTIC GEOMETRY. 309
quish mathematics, and for nearly twenty years to devote
Ms energies to physics. Important discoveries on Fresnel s
wave-surface, magnetism, spectrum-analysis were made by
Mm. But towards the close of his life he returned to his
first love, mathematics, and enriched it with new discov
eries. By considering space as made up of lines he created
a " new geometry of space." Eegarding a right line as a
curve involving four arbitrary parameters, one has the whole
system of lines in space. By connecting them by a single
relation, he got a " complex " of lines ; by connecting them
with a twofold relation, he got a " congruency " of lines. His
first researches on this subject were laid before the Eoyal
Society in 1865. His further investigations thereon appeared
in 1868 in a posthumous work entitled Neue Geometric des
Maumes gegrundet auf die Betrachtung der geraden Linie als
Eaumelement, edited by Felix Klein. Pliicker s analysis lacks
the elegance found in Lagrange, Jacobi, Hesse, and Clebsch.
For many years he had not kept up with the progress of
geometry, so that many investigations in his last work had
already received more general treatment on the part of others.
The work contained, nevertheless, much that was fresh and
original. The theory of complexes of the second degree, left
unfinished by Plucker, was continued by Felix Klein, who
greatly extended and supplemented the ideas of his master.
Ludwig Otto Hesse (1811-1874) was born at Konigsberg, and
studied at the university of his native place under Bessel,
Jacobi, Eichelot, and F. Neumann. Having taken the doctor s
degree in 1840, he became decent at Konigsberg, and in 1845
extraordinary professor there. Among his pupils at that time
were Durege, Carl Neumann, Clebsch, Kirchhoff. The Konigs
berg period was one of great activity for Hesse. Every new
discovery increased his zeal for still greater achievement.
His earliest researches were on surfaces of the second order,
310 A HISTORY OF MATHEMATICS.
and were partly synthetic. He solved the problem to construct
any tenth point of such a surface when nine points are given.
The analogous problem for a conic had been solved by Pascal
by means of the hexagram. A difficult problem confronting
mathematicians of this time was that of elimination. Pliicker
had seen that the main advantage of his special method in
analytic geometry lay in the avoidance of algebraic elimina
tion. Hesse, however, showed how by determinants to make
algebraic elimination easy. In his earlier results he was
anticipated by Sylvester, who published his dialytic method
of elimination in 1840. These advances in algebra Hesse
applied to the analytic study of curves of the third order. By
linear substitutions, he reduced a form of the third degree in
three variables to one of only four terms, and was led to an
important determinant involving the second differential coeffi
cient of a form of the third degree, called the "Hessian. 55
The " Hessian " plays a leading part in the theory of invari
ants, a subject first studied by Cayley. Hesse showed that
his determinant gives for every curve another curve, such that
the double points of the first are points on the second, or
"Hessian." Similarly for surfaces (Crelle, 1844). Many of
the most important theorems on curves of the third order are
due to Hesse. He determined the curve of the 14th order,
which passes through the 56 points of contact of the 28 bi-
tangents of a curve of the fourth order. His great memoir on
this subject (Crelle, 1855) was published at the same time as
was a paper by Steiner treating of the same subject.
Hesse s income at Konigsberg had not kept pace with his
growing reputation. Hardly was he able to support himself
and family. In 1855 he accepted a more lucrative position at
Halle, and in 1856 one at Heidelberg. Here he remained until
1868, when he accepted a position at a technic school in
Munich. 67 At Heidelberg he revised and enlarged upon his
ANALYTIC GEOMETRY. 311
previous researches, and published in 1861 Ms Vorlesungen
uber die Analytiscke Geometrie des Itaumes, insbesondere uber
Flclclien 2. Ordnung. More elementary works soon followed.
While in Heidelberg he elaborated a principle, his " Uebertra-
gungsprincip." According to this., there corresponds to every
point in a plane a pair of points in a line, and the projective
geometry of the plane can be carried back to the geometry of
points in a line.
The researches of Plucker and Hesse were continued in Eng-
land by Cay ley, Salmon, and Sylvester. It may be premised here
that among the early writers on analytical geometry in England
was James Booth (1806-1878), whose chief results are embodied
in his Treatise on Some New Geometrical Methods; and James
MacCullagh (1809-1846), who was professor of natural philos
ophy at Dublin, and made some valuable discoveries on the
theory of quadrics. The influence of these men on the
progress of geometry was insignificant, for the interchange of
scientific results between different nations was not so complete
at that time as might have been desired. In further illustra
tion of this, we mention that Chasles in France elaborated
subjects which had previously been disposed of by Steiner in
Germany, and Steiner published researches which had been
given by Cayley, Sylvester, and Salmon nearly five years
earlier. Cayley and Salmon in 1849 determined the straight
lines in a cubic surface, and studied its principal properties,
while Sylvester in 1851 discovered the pentahedron of such a
surface. Cayley extended Plueker s equations to curves of
higher singularities. Cayley s own investigations, and those
of M. Nother of Erlangen, G. H. Halphen (1844-1889) of the
Polytechnic School in Paris, De La Gournerie of Paris, A.
Brill of Tubingen, lead to the conclusion that each higher sin
gularity of a curve is equivalent to a certain number of simple
singularities, the node, the ordinary cusp, the double tangent,
312 A HISTOBY OF MATHEMATICS.
and the inflection. Sylvester studied the " twisted Cartesian,"
a curve of the fourth order. Salmon helped powerfully
towards the spreading of a knowledge of the new algebraic and
geometric methods by the publication of an excellent series of
text-books (Conic Sections, Modern Higher Algebra, Higher
Plane Curves, Geometry of Three Dimensions), which have
been placed within easy reach of German readers by a free
translation, with additions, made by Wilhelm Fiedler of the
Polytechnicurn in Zurich. The next great worker in the field
of analytic geometry was Clebsch.
Rudolf Friedrich Alfred Clebsch (1833-1872) was born at
Konigsberg in Prussia, studied at the university of that place
under Hesse, Kichelot, F. Neumann. From 1858 to 1863 he
held the chair of theoretical mechanics at the Polytechnicum
in Carlsruhe. The study of Salmon s works led him into
algebra and geometry. In 1863 he accepted a position at the
University of Giesen, where he worked in conjunction with
Paul Gordan (now of Erlangen). In 1868 Clebsch went to
Gottingen, and remained there until his death. He worked
successively at the following subjects : Mathematical physics,
the calculus of variations and partial differential equations of
the first order, the general theory of curves and surfaces,
Abelian functions and their use in geometry, the theory of
invariants, and " Flachenabbildung." He proved theorems
on the pentahedron enunciated by Sylvester and Steiner ; he
made systematic use of "deficiency" (Geschlecht) as a funda
mental principle in the classification of algebraic curves. The
notion of deficiency was known before him to Abel and Eie-
mann. At the beginning of his career, Clebsch had shown
how elliptic functions could be advantageously applied to
Malfatti s problem. The idea involved therein, viz. the use
of higher transcendentals in the study of geometry, led him
to his greatest discoveries. Not only did he apply Abelian
ANALYTIC GEOMETRY. 313
functions to geometry, but conversely, he drew geometry into
the service of Abelian functions.
Clebsch made liberal use of determinants. His study of
curves and surfaces began with the determination of the points
of contact of lines which meet a surface in four consecutive
points. Salmon had proved that these points lie on the inter
section of the surface with a derived surface of the degree
H w _24, but his solution was given in inconvenient form.
Clebsch s investigation thereon is a most beautiful piece of
analysis.
The representation of one surface upon another (Fldchenab-
bildung), so that they have a (1, 1) correspondence, was
thoroughly studied for the first time by Clebsch. The repre
sentation of a sphere on a plane is an old problem which
drew the attention of Ptolemaeus, Gerard Mercator, Lambert,
Gauss, Lagrange. Its importance in the construction of maps
is obvious. Gauss was the first to represent a surface upon
another with a view of more easily arriving at its properties.
Plucker, Chasles, Cayley, thus represented on a plane the
geometry of quadrie surfaces ; Clebsch and Cremona, that of
cubic surfaces. Other surfaces have been studied in the same
way by recent writers, particularly M. Mother of Erlangen,
Armenante, Felix Klein, Korndorfer, Caporali, H. G. Zeuthen
of Copenhagen. A fundamental question which has as yet
received only a partial answer is this : What surfaces can be
represented by a (1, 1) correspondence upon a given surface ?
This and the analogous question for curves was studied by
Clebsch. Higher correspondences between surfaces have been
investigated by Cayley and ISTother. The theory of surfaces
has been studied also by Joseph Alfred Serret (1819-1885), pro
fessor at the Sorbonne in Paris, Jean Gaston Darboux of Paris,
John Casey of Dublin (died 1891), W. R. W. Roberts of Dub
lin, H. Scliroter (1829-1892) of Breslau. Surfaces of the
314 A HISTORY OF MATHEMATICS.
fourth, order were investigated by Kummer, and EresnePs
wave-surface, studied by Hamilton, is a particular case of
Kummer s quartic surface, with sixteen canonical points and
sixteen singular tangent planes. 56
The infinitesimal calculus was first applied to the determi
nation of the measure of curvature of surfaces by Lagrange,
Euler, and Meunier (1754-1793) of Paris. Then followed the
researches of Monge and Dupin, but they were eclipsed by
the work of Gauss, who disposed of this difficult subject in a
way that opened new vistas to geometricians. His treat
ment is embodied in the Disquisitiones generales circa super
ficies curvas (1827) and Vhtersuchungen uber gegenstdnde der
Jidheren Geodasie of 1843 and 1846. He defined the measure
of curvature at a point to be the reciprocal of the product
of the two principal radii of curvature at that point. From
this flows the theorem of Johann August Grunert (1797-1872 ;
professor in Greifswald), that the arithmetical mean of the
radii of curvature of all normal sections through a point is the
radius of a sphere which has the same measure of curvature
as has the surface at that point. Gauss s deduction of the
formula of curvature was simplified through the use of deter
minants by Heinrich Ricliard Baltzer (1818-1887) of Giessen. 69
Gauss obtained an interesting theorem that if one surface be
developed (abgewickelt) upon another, the measure of curva
ture remains unaltered at each point. The question whether
two surfaces having the same curvature in corresponding
points can be unwound, one upon the other, was answered
by F- Minding in the affirmative only when the curvature is
constant. The case of variable curvature is difficult, and was
studied by Minding, J. Liouville (1806-1882) of the Poly
technic School in Paris, Ossian Bonnet of Paris (died 1892).
Gauss s measure of curvature, expressed a$ a function of cur
vilinear co-ordinates, gave an impetus to the study of differ-
ALGEBRA. 315
ential-invariants, or differential-parameters, which have been
investigated by Jaeobi, C. Neumann, Sir James Cockle,
Halphen, and elaborated into a general theory by Beltrami,
S. Lie, and others. Beltrami showed also the connection
between the measure of curvature and the geometric axioms.
Various researches have been brought under the head of
"analysis situs." The subject was first investigated by
Leibniz, and was later treated by Gauss, whose theory of
knots (VerschUngungen) has been employed recently by J. B.
Listing, 0. Simony, E. Dingeldey, and others in their "topo-
logic studies." Tait was led to the study of knots by Sir
William. Thomson s theory of vortex atoms. In the hands
of Eiemann the analysis situs had for its object the deter
mination of what remains unchanged under transformations
brought about by a combination of infinitesimal distortions.
In continuation of his work, "Walter Dyck of Munich wrote on
the analysis situs of three-dimensional spaces.
Of geometrical text-books not yet mentioned, reference
should be made to Alfred Clebsch s Vorlesungen uber Geome-
trie, edited by Ferdinand Lindemann, now of Munich ; Frost s
Solid Geometry; Durege s Ebene Ourven dritter Ordnung.
ALGEBBA.
The progress of algebra in recent times may be considered
under three principal heads : the study of fundamental laws
and the birth of new algebras, the growth of the theory of
equations, and the development of what is called modern
higher algebra.
We have already spoken of George Peacock and D. F.
Gregory in connection with the fundamental laws of algebra.
Much was done in this line by De Morgan.
316 A HISTORY OF MATHEMATICS.
Augustus De Morgan (1806-1871) was bom at Madura (Ma
dras), and educated at Trinity College, Cambridge. His scru
ples about the doctrines of the established church prevented
him from, proceeding to the M.A. degree, and from sitting
for a fellowship. In 1828 he became professor at the newly
established University of London, and taught there until
1867, except for five years, from 1831-1835. De Morgan was
a unique, manly character, and pre-eminent as a teacher. The
value of his original work lies not so much in increasing our
stock of mathematical knowledge as in putting it all upon a
thoroughly logical basis. He felt keenly the lack of close
reasoning in mathematics as he received it. He said once :
" We know that mathematicians care no more for logic than
logicians for mathematics. The two eyes of exact science are
mathematics and logic : the mathematical sect puts out the
logical eye, the logical sect puts out the mathematical eye;
each believing that it can see better with one eye than with
two." De Morgan saw with both eyes. He analysed logic
mathematically, and studied the logical analysis of the laws,
symbols, and operations of mathematics ; he wrote a Formal
Logic as well as a Double Algebra^ and corresponded both with
Sir William Hamilton, the metaphysician, and Sir William
Rowan Hamilton, the mathematician. Few contemporaries
were as profoundly read in the history of mathematics as
was De Morgan. No subject was too insignificant to receive
Ms attention. The authorship of "Cocker s Arithmetic" and
the work of circle-squarers was investigated as minutely as was
the history of the invention of the calculus. Numerous arti
cles of his lie scattered in the volumes of the Penny and Eng
lish Cydopcedias. His Differential Calculus, 1842, is still a
standard work, and contains much that is original with the
author. For the Encyclopaedia Metropolitans he wrote on the
calculus of functions (giving principles of symbolic reasoning)
ALGEBRA. 817
and on the theory of probability. Celebrated is his Budget of
Paradoxes, 1872. He published memoirs " On the Foundation
of Algebra" (Trans, of Gam. Phil Soc., 1841, 1842, 1844, and
1847).
In Germany symbolical algebra was studied by Martin Ohm,
who wrote a System der Mathematik in 1822. The ideas of
Peacock and Be Morgan recognise the possibility of algebras
which differ from ordinary algebra. Such algebras were
indeed not slow in forthcoming, but, like non-Euclidean
geometry, some of them were slow in finding recognition.
This is true of Grassmann s, Bellavitis s, and Peirce s dis
coveries, but Hamilton s quaternions met with immediate
appreciation in England. These algebras offer a geometrical
interpretation of imaginaries. During the times of Descartes,
Newton, and Euler, we have seen the negative and the imagi
nary, V 1, accepted as numbers, but the latter was still
regarded as an algebraic fiction. The first to give it a geomet
ric picture, analogous to the geometric interpretation of the
negative, was H. Kuhn, a teacher in Danzig, in a publication of
1750-1751. He represented aV^l by a line perpendicular
to the line a, and equal to a in length, and construed V 1 as
the mean proportional between + 1 and 1. This same idea
was developed further, so as to give a geometric interpretation
of a-f V^, by Jean-Robert Argand (1768- ?) of Geneva,
in a remarkable Essai (1806) . 70 The writings of Kuhn and
Argand were little noticed, and it remained for Gauss to break
down the last opposition to the imaginary. He introduced i as
an independent unit co-ordinate to 1, and a + ft> as a " complex
number." The connection between complex numbers and
points on a plane, though artificial, constituted a powerful
aid in the further study of symbolic algebra. The mind
required a visual representation to aid it. The notion of
what we now call vectors was growing upon mathematicians,
318 A HISTORY OF MATHEMATICS.
and tlie geometric addition of vectors in space was discovered
independently by Hamilton, G-rassmann, and others, about the
same time.
William Rowan Hamilton (1805-1865) was born of Scotch
parents in Dublin. His early education, carried on at home,
was mainly in languages. At the age of thirteen he is said to
have been familiar with as many languages as he had lived
years. About this time he caine across a copy of Newton s
Universal Arithmetic. After reading that, he took up succes
sively analytical geometry, the calculus, Newton s Principia,
Laplace s Mecanique Celeste. At the age of eighteen he
published a paper correcting a mistake in Laplace s work. In
1824 he entered Trinity College, Dublin, and in 1827, while he
was still an undergraduate, he was appointed to the chair of
astronomy. His early papers were on optics. In 1832 he
predicted conical refraction, a discovery by aid of mathe
matics which ranks with the discovery of Neptune by
Le Yerrier and Adams. Then followed papers on the Prin
ciple of Varying Action (1827) and a general method of
dynamics (1834-1835). He wrote also on the solution of
equations of the fifth degree, the hodograph, fluctuating
functions, the numerical solution of differential equations.
The capital discovery of Hamilton is his quaternions, in
which his study of algebra culminated. In 1835 he published
in the Transactions of the Royal Irish Academy his Theory of
Algebraic Couples. He regarded algebra " as being no mere
art, nor language, nor primarily a science of quantity, but
rather as the science of order of progression." Time appeared
to him as the picture of such a progression. Hence his defini
tion of algebra as "the science of pure time." It was the
subject of years meditation for him to determine what he
should regard as the product of each pair of a system of per
pendicular directed lines. At last, on the 16th of October,
ALGEBRA. 319
1843, while walking witii Ms wife one evening, along the
Koyal Canal in Dublin, the discovery of quaternions flashed
upon him, and he then engraved with his knife on a stone in
Brougham Bridge the fundamental formula i 2 =/ = If = ijJc =
1. At the general meeting of the Irish Academy, a month
later, he made the first communication on quaternions. An
account of the discovery was given the following year in the
Philosophical Magazine. Hamilton displayed wonderful fer
tility in their development. His Lectures on Quaternions,
delivered in Dublin, were printed in 1852. His Elements of
Quaternions appeared in 1866. Quaternions were greatly
admired in England from the start, but on the Continent they
received less attenttion. P. G. Tait s Elementary Treatise
helped powerfully to spread a knowledge of them in England.
Cay ley, Clifford, and Tait advanced the subject somewhat by
original contributions. But there has been little progress in
recent years, except that made by Sylvester in tjjp solution of
quaternion equations, nor has the application of quaternions
to physics been as extended as was predicted. The change
in notation made in France by Houel and by Laisant has been
considered in England as a wrong step, but the true cause for
the lack of progress is perhaps more deep-seated. There is
indeed great doubt as to whether the quaternionic product can
claim a necessary and fundamental place in a system of vector
analysis. Physicists claim that there is a loss of naturalness
in taking the square of a vector to be negative. In order to
meet more adequately their wants, J. Wl Gfibbs of Yale Uni
versity and A. Macfarlane of the University of Texas, have
each suggested an algebra of vectors with a new notation.
Each gives a definition of his own for the product of two
vectors, but in such a way that the square of a vector is
positive. A third system of vector analysis has been used by
Oliver Heaviside in his electrical researches.
320 A HISTORY OF MATHEMATICS.
Hermann Grassmann (1809-1877) was bora at Stettin,
attended a gymnasium at Ms native place (where his father
was teacher of mathematics and physics) , and stndied theology
in Berlin for three years. In 1834 he succeeded Steiner as
teacher of mathematics in an industrial school in Berlin, but
returned to Stettin in 1836 to assume the duties of teacher of
mathematics, the sciences, and of religion in a school there. 71
Up to this time his knowledge of mathematics was pretty
much confined to what he had learned from his father, who
had written two books on "Baumlehre" and " Grossenlehre."
But now he made his acquaintance with the works of Lacroix,
Lagrange, and Laplace. He noticed that Laplace s results
could be reached in a shorter way by some new ideas advanced
in his father s books, and he proceeded to elaborate this
abridged method, and to apply it in the study of tides. He
was thus led to a new geometric analysis. In 1840 he had
made consic|jrable progress in its development, but a new
book of Schleiermacher drew him again to theology. In
1842 he resumed mathematical research, and becoming thor
oughly convinced of the importance of his new analysis,
decided to devote himself to it. It now became his ambition
to secure a mathematical chair at a university, but in this he
never succeeded. In 1844 appeared his great classical work,
the Lineale Ausdehnungslelire, which was full of new and
strange matter, and so general, abstract, and out of fashion in
its mode of exposition, that it could hardly have had less
influence on European mathematics during its first twenty
years, had it been published in China. Gauss, Grunert, and
Mobius glanced over it, praised it, but complained of the
strange terminology and its " philosophische Allgemeinheit."
Eight years afterwards, Bretschneider of Gotha was said to be
the only man who had read it through. An article in Crelle s
Journal, in which Grassmann eclipsed the geometers of that
ALGEBRA. 321
time by constructing, with, aid of Ms method, geometrically
any algebraic curve, remained again unnoticed. Need we mar
vel if G-rassinann turned Ms attention to other subjects, to
Schleiermacher s pMlosophy, to polities, to philology ? Still,
articles by him continued to appear in Crelle s Journal, and in
1862 came out the second part of Ms Ausdehnungslehre. It
was intended to show better than the first part the broad
scope of the Ausdehnungslehre, by considering not only geo
metric applications, but by treating also of algebraic functions,
infinite series, and the differential and integral calculus. But
the second part was no more appreciated than the first. At the
age of fifty-three, this wonderful man, with heavy heart, gave
up mathematics, and directed his energies to the study of Sans
krit, achieving in philology results which were better appreci
ated, and which vie in splendour with those in mathematics.
Common to the Ausdehnungslehre and to quaternions are geo
metric addition, the function of two vectors represented in qua
ternions by Saft and Fa/3, and the linear vector functions. The
quaternion is peculiar to Hamilton, while with Grassmann we
find in addition to the algebra of vectors a geometrical algebra
of wide application, and resembling Mobius s Barycentrische
Calculj in which the point is the fundamental element. Grass
mann developed the idea of the "external product," the "inter
nal product," and the " open product." The last we now call
a matrix. His Ausdehnungslehre has very great extension,
having no limitation to any particular number of dimen
sions. Only in recent years has the wonderful richness of
his discoveries begun to be appreciated. A second edition of
the Ausdehnungslehre of 1844 was printed in 1877. C. S.
Peirce gave a representation of Grassmann s system in the
logical notation, and E. W. Hyde of the University of Cin
cinnati wrote the first text-book on Grassmann s calculus in
the English language.
322 A HISTORY OF MATHEMATICS.
Discoveries of less value, which in part covered those of
Grassmann and Hamilton, were made by Saint- Venant (1797-
1886), who described the multiplication of vectors, and the
addition of vectors and oriented areas; by Cauchy; whose
"clefs algebriques " were units subject to combinatorial mul
tiplication, and were applied by the author to the theory of
elimination in the same way as had been done earlier by
Grassmann; by Justus Bellavitis (1803-1880), who published
in 1835 and 1837 in the Annali delle Scienze his calculus of
sequipollences. Bellavitis, for many years professor at Padua,
was a self-taught mathematician of much power, who in his
thirty-eighth year laid down a city office in his native place,
Bassano, that he might give his time to science. 65
The first impression of G-rassmann s ideas is marked in the
writings of Hermann Hankel (1839-1873), who published in
1867 his Vorlesungen uber die Complexen Zahlen. Hankel,
then decent in Leipzig, had been in correspondence with
Grassraann. The "alternate numbers" of Hankel are sub
ject to his law of combinatorial multiplication. In consider
ing the foundations of algebra Hankel affirms the principle of
the permanence of formal laws previously enunciated incom
pletely by Peacock. Hankel was a close student of mathe
matical history, and left behind an unfinished work thereon.
Before his death he was professor at Tubingen. His Com-
plexe Zahlen was at first little read, and we must turn to
Victor ScMegel of Hagen as the successful interpreter of Grass
mann. Schlegel was at one time a young colleague of Grass-
maun at the Marienstifts-Gyrrmasiuin in Stettin. Encouraged
by Clebsch, Schlegel wrote a System der Baumlehre which
explained the essential conceptions and operations of the
Ausdehnungslehre.
Multiple algebra was powerfully advanced by Peirce, whose
theory is not geometrical, as are those of Hamilton and Grass-
ALGEBBA.
mann. Benjamin Peirce (1809-1880) was born at Salem, Mass.,
and graduated at Harvard College, having as undergraduate
carried the study of mathematics far beyond the limits of the
college course. 2 When Bowditch was preparing his transla
tion and commentary of the Mecanique Ctteste, young Peirce
helped in reading the proof-sheets. He was made professor at
Harvard in 1833, a position which he retained until his death.
Eor some years he was in charge of the Nautical Almanac
and superintendent of the United States Coast Survey. He
published a series of college text-books on mathematics, an
Analytical Mechanics, 1855, and calculated, together with Sears
C. Walker of Washington, the orbit of Neptune. Profound
are his researches on Linear Associative Algebra. The first of
several papers thereon was read at the first meeting of the
American Association for the Advancement of Science in 1864.
Lithographed copies of a memoir were distributed among friends
in 1870, but so small seemed to be the interest taken in this
subject that the memoir was not printed until 1881 (Am. Jour.
Matli.y Vol. IV., Eo. 2). Peirce works out the multiplication
tables, first of single algebras, then of double algebras, and so
on up to sextuple, making in all 162 algebras, which he shows
to be possible on the consideration of symbols A, B, etc.,
which are linear functions of a determinate number of letters
or units i, j, Jc, I, etc., with coefficients which are ordinary ana
lytical magnitudes, real or imaginary, the letters i, j, etc.,
being such that every binary combination ft, ij, ji, etc., is equal
to a linear function of the letters, but under the restriction of
satisfying the associative law. 56 Charles S. Peirce, a son of
Benjamin Peirce, and one of the foremost writers on mathe
matical logic, showed that these algebras were all defective
forms of quadrate algebras which he had previously discovered
by logical analysis, and for which he had devised a simple
notation. Of these quadrate algebras quaternions is a simple
324 A HISTORY OF MATHEMATICS.
example ; nonions is another. C. S. Peirce showed that of all
linear associative algebras there are only three in whieh divis
ion is -unambiguous. These are ordinary single algebra,
ordinary double algebra, and quaternions, from which the
imaginary scalar is excluded. He showed that his father s
algebras are operational and matricular. Lectures on multiple
algebra were delivered by J. J. Sylvester at the Johns Hopkins
University, and published in various journals. They treat
largely of the algebra of matrices. The theory of matrices
was developed as early as 1858 by Cayley in an important
memoir which, in the opinion of Sylvester, ushered in the
reign of Algebra the Second. Clifford, Sylvester, H. Taber,
C. H. Chapman, carried the investigations much further. The
originator of matrices is really Hamilton, but his theory, pub
lished in his Lectures on Quaternions, is less general than that
of Cayley, The latter makes no reference to Hamilton.
The theory of determinants 73 was studied by Hoene Wronski
in Italy and J. Binet in Prance ; but they were forestalled by
the great master of this subject, Cauchy. In a paper (Jour,
de Vecole Polyt., IX., 16) Cauchy developed several general
theorems. He introduced the name determinant, a term
previously used by Gauss in the functions considered by him.
In 1826 Jacobi began using this calculus, and he gave brilliant
proof of its power. In 1841 he wrote extended memoirs on
determinants in Qrelle s Journal, which rendered the theory
easily accessible. In England the study of linear transforma
tions of quantics gave a powerful impulse. Cayley developed
skew-determinants and Pfafftans, and introduced the use of
determinant brackets, or the familiar pair of upright lines.
More recent researches on determinants appertain to special
forms. "Continuants" are due to Sylvester; "alternants,"
originated by Cauchy, have been developed by Jacobi, N". Trudi,
H. Nagelbach, and G. Garbieri ; " axisymmetric determinants/
ALGEBBA. 325
first used by Jacobi, have been studied by V. A. Lebesgue,
Sylvester, and Hesse ; " eirculants " are due to E. Catalan of
Liege, W. Spottiswoode (1825-1883), J. W. L. Glaisher, and
E. F. Scott; for " centre-symmetric determinants" we are
indebted to G. Zehfuss. E. B. Christoffel of Strassburg and
G. Frobenius discovered the properties of Wronskians," first
used by WronskL V. Nachreiner and S. Giinther, both of
Munich, pointed out relations between determinants and con
tinued fractions ; Scott uses HankePs alternate numbers in his
treatise. Text-books on determinants were written by Spot
tiswoode (1851), Brioschi (1854), Baltzer (1857), Gunther
(1875), Dostor (1877), Scott (1880), Muir (1882), Hanus
(1886).
Modern higher algebra is especially occupied with the
theory of linear transformations. Its development is mainly
the work of Cayley and Sylvester.
Arthur Cayley, born at Eichmond, in Surrey, in 1821^ was
educated at Trinity College, Cambridge. 74 He came out Senior
Wrangler in 1842. He then devoted some years to the study
and practice of law. On the foundation of the Sadlerian pro
fessorship at Cambridge, he accepted the offer of that chair,
thus giving up a profession promising wealth for a very
modest provision, but which would enable him to give all his
time to mathematics. Cayley began his mathematical publi
cations in the Cambridge Mathematical Journal while he was
still an undergraduate. Some of his most brilliant discoveries
were made during the time of his legal practice. There is
hardly any subject in pure mathematics which the genius of
Cayley has not enriched, but most important is his creation
of a new branch of analysis by his theory of invariants.
Germs of the principle of invariants are found in the writ
ings of Lagrange, Gauss, and particularly of Boole, who
.showed, in 1841, that invariance is a property of discrimi-
326 A H1STOBY OF MATHEMATICS.
nants generally, and who applied it to the theory of orthogonal
substitution. Cayley set himself the problem to determine
a priori what functions of the coefficients of a given equation
possess this property of invariance, and found, to begin with,
in 1845, that the so-called " hyper-determinants " possessed it.
Boole made a number of additional discoveries. Then Syl
vester began his papers in the Cambridge and Dublin Mathe
matical Journal on the Calculus of Forms. After this,
discoveries followed in rapid succession. At that time Cay-
ley and Sylvester were both residents of London, and they
stimulated each other by frequent oral communications. It
has often been difficult to determine how much really belongs
to each.
James Joseph Sylvester was born in London in 1814, and
educated at St. Johns College, Cambridge. He came out
Second Wrangler in 1837. His Jewish origin incapacitated
him from taking a degree. In 1846 he became a student at
the Inner Temple, and was called to the bar in 1850. He
became professor of natural philosophy at University College,
London ; then, successively, professor of mathematics at the
University of Virginia, at the Eoyal Military Academy in
Woolwich, at the Johns Hopkins University in Baltimore,
and is, since 1883, professor of geometry at Oxford. His
first printed paper was on Fresnel s optic theory, 1837. Then
followed his researches on invariants, the theory of equations,
theory of partitions, multiple algebra, the theory of numbers,
and other subjects mentioned elsewhere. About 1874 he took
part in the development of the geometrical theory of link-
work movements, originated by the beautiful discovery of
A. Peaucellier, Capitaine du Genie & Nice (published in
Nouvelles Annales, 1864 and 1873), and made the subject
of close study by A. B. Kempe. To Sylvester is ascribed the
general statement of the theory of contravariants, the dis-
ALGEBBA. 327
covery of the partial differential equations satisfied by the
invariants and oo variants of binary qualities, and the subject
of mixed concomitants. In the American Journal of Mathe
matics are memoirs on binary and ternary quantics, elaborated
partly with aid of F. Franklin, now professor at the Johns
Hopkins University. At Oxford, Sylvester has opened up a
new subject, the theory of reciprocals, treating of the func
tions of a dependent variable y and the functions of its differ
ential coefficients in regard to x, which remain unaltered by
the interchange of x and y. This theory is more general than
one on differential invariants by Halphen (1878), and has
been developed further by J. Hammond of Oxford, McMahon
of Woolwich, A. E. Forsyth of Cambridge, and others. Syl
vester playfully lays claim to the appellation of the Mathe
matical Adam, for the many names he has introduced into
mathematics. Thus the terms invariant, discriminant, Hessian,
Jacobian, are his.
The great theory of invariants, developed in England mainly
by Cayley and Sylvester, came to be studied earnestly in Ger
many, France, and Italy. One of the earliest in the field was
Siegfried Heinrich Aronhold (1819-1884), who demonstrated
the existence of invariants, S and T, of the ternary cubic.
Hermite discovered evectants and the theorem of reciprocity
named after him. Paul Gordan showed, with the aid of
symbolic methods, that the number of distinct forms for a
binary quantic is finite. Clebsch proved this to be true for
quantics with any number of variables. A very much simpler
proof of this was given in 1891, by David Hilbert of Konigs-
berg. In Italy, F. Brioschi of Milan and Fa& de Bruno
(1825-1888) contributed to the theory of invariants, the
latter writing a text-book on binary forms, which ranks by
the side of Salmon s treatise and those of Clebsch and
Gordan. Among other writers on invariants are E. E. Chris-
328 A HISTOBY OF MATHEMATICS.
toffel, Wilhelm Eiedler, P. A. McMahon, J. W. L. G-laisher of
Cambridge, Emory McClintock of N ew York. McMalion dis
covered that the theory of semi-invariants is a part of that of
symmetric functions. The modern higher algebra has reached
out and indissolubly connected itself with several other
branches of mathematics geometry , calculus of variations,
mechanics. Clebsch extended the theory of binary forms to
ternary, and applied the results to geometry. Clebsch, Klein,
Weierstrass, Burckhardt, and Bianchi have used the theory of
invariants in hyperelliptic and Abelian functions.
In the theory of equations Lagrange, Argand, and Gauss
furnished proof to the important theorem that every algebraic
equation has a real or a complex root. Abel proved rigorously
that the general algebraic equation of the fifth or of higher
degrees cannot be solved by radicals (Crelle, I., 1826), A modi
fication of Abel s proof was given by Wantzel. Before Abel,
an Italian physician, Paolo Euffini (1765-1822), had printed
proofs of the insolvability, which were criticised by his
countryman Malfatti. Though inconclusive, Buffings papers
are remarkable as containing anticipations of Cauchy s theory
of groups. 76 A transcendental solution of the quintic involving
elliptic integrals was given by Hermite (Compt Rend., 1858,
1865, 1866). After Hermite s first publication, Kronecker, in
1858, in a letter to Hermite, gave a second solution in which
was obtained a simple resolvent of the sixth degree. Jerrard, in
his Mathematical Researches (1832-1835), reduced the quintic
to the trinomial form by an extension of the method of
Tschirnhausen. This important reduction had been effected
as early as 1786 by E. 8. Bring, a Swede, and brought out
in a publication of the University of Lund. Jerrard, like
Tschirnhausen, believed that his method furnished a general
algebraic solution of equations of any degree. In 1836 Wil
liam B. Hamilton made a report on the validity of Jerrard s
ALGEBKA. 329
method, and showed that by his process the quintic could be
transformed to any one of the four trinomial forms. Ham
ilton denned the limits of its applicability to higher equations.
Sylvester investigated this question, What is the lowest degree
an equation can have in order that it may admit of being
deprived of i consecutive terms by aid of equations not higher
than fth degree. He carried the investigation as far as i = 8,
and was led to a series of numbers which he named "Hamilton s
numbers." A transformation of equal importance to Jerrard s
is that of Sylvester, who expressed the quintic as the sum of
three fifth-powers. The covariants and invariants of higher
equations have been studied much in recent years.
Abel s proof that higher equations cannot always be solved
algebraically led to the inquiry as to what equations of a given
degree can be solved by radicals. Such equations are the
ones discussed by Gauss in considering the division of the
circle. Abel advanced one step further by proving that an
irreducible equation can always be solved in radicals, if, of
two of its roots, the one can be expressed rationally in terms
of the other, provided that the degree of the equation is
prime ; if it is not prime, then the solution depends upon
that of equations of lower degree. Through geometrical con
siderations, Hesse came upon algebraically solvable equations
of the ninth degree, not included in the previous groups.
The subject was powerfully advanced in Paris by the youthful
Evariste Galois (born, 1811; killed in a duel, 1832), who
introduced the notion of a group of substitutions. To him are
due also some valuable results in relation to another set of
equations, presenting themselves in the theory of elliptic
functions, viz. the modular equations. G-alois s labours gave
birth to the important theory of substitutions, which has been
greatly advanced by G. Jordan of Paris, J. A. Serret (1819-
1885) of the Sorbonne in Paris, I. Kronecker (1823-1891) of
380 A HISTORY OF MATHEMATICS.
Berlin, Klein of Gottingen, M. Mother of Erlangen, 0.
Herniite of Paris, A. Capelli of Naples, L. Sylow of Fried-
richshald, E. Netto of Giessen. Eetto s book, the Substitu-
tionstheorie, has been translated into English by F. N. Cole of
the University of Michigan, who contributed to the theory.
A simple group of 504 substitutions of nine letters, discovered
by Cole, has been shown by E. H. Moor of the University of
Chicago to belong to a doubly-infinite system of simple groups.
The theory of substitutions has important applications in the
theory of differential equations. Kronecker published, in 1882,
his Gf-rundzuge einer Arithmetischen Theorie der Algebraischen
Gfrossen.
Since Fourier and Budan, the solution of numerical equa
tions has been advanced by W. Gr. Homer of Bath, who gave
an improved method of approximation (Philosophical Transac
tions, 1819). Jacques Charles Francois Sturm (1803-1855), a
native of Geneva, Switzerland, and the successor of Poisson
in the chair of mechanics at the Sorbonne, published -in 1829
his celebrated theorem determining the number and situation
of roots of an equation comprised between given limits. Sturm
tells us that his theorem stared him in the face in the midst
of some mechanical investigations connected with the motion
of a compound pendulum. 77 This theorem, and Homer s
method, offer together sure and ready means of finding the
real roots of a numerical equation.
The symmetric functions of the sums of powers of the roots
of an equation, studied by Newton and Waring, was considered
more recently by Gauss, Cayley, Sylvester, BrioschL Cayley
gives rules for the " weight " and " order " of symmetric func
tions.
The theory of elimination was greatly advanced by Sylves
ter, Cayley, Salmon, Jacobi, Hesse, Cauchy, Brioschi, and
Gordan. Sylvester gave the dialytic method (Philosophical
ANALYSIS. 331
Magazine, 1840), and .in 1852 established a theorem relating
to the expression of an eliminant as a determinant. Cayley
made a new statement of Bezout s method of elimination and
established a general theory of elimination (1852).
ANALYSIS.
Under this head we find it convenient to consider the sub
jects of the differential and integral calculus, the calculus of
variations, infinite series, probability, and differential equa
tions. Prominent in the development of these subjects was
Cauchy.
Augtistin-Louis Cauchy 78 (1789-1857) was born in Paris,
and received his early education from his father. Lagrange
and Laplace, with whom the father came in frequent contact,
foretold the future greatness of the young boy. At the
ICcole Cent rale du Pantheon he excelled in ancient classical
studies. In 1805 he entered the Polytechnic School, and two
years later the cole des Ponts et Chaussees. Cauchy left for
Cherbourg in 1810, in the capacity of engineer. Laplace s
M&camque C&leste and Lagrange 7 s Fonctions Analytiques were
among his book companions there. Considerations of health
induced him to return to Paris after three years. Yielding to
the persuasions of Lagrange and Laplace, he renounced engi
neering in favour of pure science. We find him next holding
a professorship at the Polytechnic School. On the expulsion
of Charles X., and the accession to the throne of Louis Philippe
in 1830, Cauchy, being exceedingly conscientious, found him
self unable to take the oath demanded of him. Being, in
consequence, deprived of his positions, he went into volun
tary exile. At TMbourg in Switzerland, Cauchy resumed his
studies, and in 1831 was induced by the king of Piedmont to
332 A HISTOEY OF MATHEMATICS.
accept the chair of mathematical physics, especially created
for him at the university of Turin. In 1833 he obeyed the
call of his exiled king, Charles X., to undertake the education
of a grandson, the Duke of Bordeaux. This gave Cauchy an
opportunity to visit various parts of Europe, and to learn how
extensively his works were being read. Charles X. bestowed
upon him the title of Earon. On his return to Paris in 1838,
a chair in the College de France was offered to him, but the
oath demanded of him prevented his acceptance. He was
nominated member of the Bureau of Longitude, but declared
ineligible by the ruling power. During the political events of
1848 the oath was suspended, and Cauchy at last became
professor at the Polytechnic School. On the establishment of
the second empire, the oath was re-instated, but Cauchy and
Arago were exempt from it. Cauchy was a man of great
piety, and in two of his publications staunchly defended the
Jesuits.
Cauchy was a prolific and profound mathematician. By a
prompt publication of his results, and the preparation of
standard text-books, he -exercised a more immediate and
beneficial influence upon the great mass of mathematicians
than any contemporary writer. He was one of the leaders
in infusing rigour into analysis. His researches extended
over the field of series, of imaginaries, theory of numbers,
differential equations, theory of substitutions, theory of func
tions, determinants, mathematical astronomy, light, elasticity,
etc., covering pretty much the whole realm of mathematics,
pure and applied.
Encouraged by Laplace and Poisson, Cauchy published in
1821 his Cours d } Analyse de VEcole Royale Polytecfmique, a
work of great merit. Had it been studied more diligently by
writers of text-books in England and the United States, many
a lax and loose method of analysis hardly as yet eradicated
ANALYSIS. 383
from elementary text-books would have been discarded over
half a century ago. Cauchy was the first to publish a
rigorous proof of Taylor s theorem. He greatly improved
the exposition of fundamental principles of the differential
calculus by his mode of considering limits and his new theory
on the continuity of functions. The method of Cauehy and
Duhamel was accepted with favour by Houel and others. In
England special attention to the clear exposition of funda
mental principles was given by De Morgan. Recent American
treatises on the calculus introduce time as an independent vari
able, and the allied notions of velocity and acceleration. thus
virtually returning to the method of fluxions.
Cauchy made some researches on the calculus of variations.
This subject is now in its essential principles the same as
when it came from the hands of Lagrange. Eecent studies
pertain to the variation of a double integral when the limits
are also variable, and to variations of multiple integrals in
general. Memoirs were published by Gauss in 1829, Poisson
in 1831, and Ostrogradsky of St. Petersburg in 1834, without,
however, determining in a general manner the number and
form of the equations which must subsist at the limits in case
of a double or triple integral. In 1837 Jacobi published a
memoir, showing that the difficult integrations demanded by
the discussion of the second variation, by which the existence
of a maximum or minimum can be ascertained, are included
in the integrations of the first variation, and thus are super
fluous. This important theorem, presented with great brevity
by Jacobi, was elucidated and extended by V. A. Lebesgue,
C. E. Delaunay, Eisenlohr, S. Spitzer, Hesse, and Clebsch. An
important memoir by Sarrus on the question of determining
the limiting equations which must be combined with the
indefinite equations in order to determine completely the
maxima and minima of multiple integrals, was awarded a
334 A HISTOKY OF MATHEMATICS.
prize by the French Academy in 1845, honourable mention
being made of a paper by Delaunay. Sarrus s method was
simplified by Cauchy. In 1852 G. Mainardi attempted to
exhibit a new method of discriminating maxima and minima,
and extended Jacobi s theorem to double integrals. Mainardi
and F. Brioschi showed the value of determinants in exhibit
ing the terms of the second variation. In 1861 Isaac Tod-
hunter (1820-1884) of St. John s College, Cambridge, published
his valuable work on the History of the Progress of the Calculus
of Variations, which contains researches of his own. In 1866
he published a most important research, developing the theory
of discontinuous solutions (discussed in particular cases by
Legendre), and doing for this subject what Sarrus had done
for multiple integrals.
The following are the more important authors of systematic
treatises on the calculus of variations, and the dates of publi
cation: Eobert Woodhouse, Fellow of Caius College, Cam
bridge, 1810 ; Richard Abbatt in London, 1837 ; John Hewitt
Jellett (1817-1888), once Provost of Trinity College, Dublin,
1850 5 G. W. Strauch in Zurich, 1849 ; Moigno and Lindelof,
1861; Lewis Buffett Carll of Flushing in New York, 1881.
The lectures on definite integrals, delivered by Dirichlet in
1858, have been elaborated into a standard work by G. F.
Meyer. The subject has been treated most exhaustively by
D. Bierens de Haan of Leiden in his Hxposd de la theorie des
integrals d&finieSj Amsterdam, 1862.
The history of infinite series illustrates vividly the salient
feature of the new era which analysis entered upon during the
first quarter of this century. Newton and Leibniz felt the
necessity of inquiring into the convergence of infinite series,
but they had no proper criteria, excepting the test advanced
by Leibniz for alternating series. By Euler and his contem
poraries the formal treatment of series was greatly extended,
ANALYSIS. 335
while tlie necessity for determining the convergence was gen
erally lost sight of. Euler reached some very pretty results
on infinite series, now well known, and also some very
absurd results, now quite forgotten. The faults of Ms time
found their culmination in the Combinatorial School in Ger
many, which has now passed into deserved oblivion. At the
beginning of the period now under consideration, the doubtful,
or plainly absurd, results obtained from infinite series stimu
lated prof ounder inquiries into the validity of operations with
them. Their actual contents came to be the primary, form a
secondary, consideration. The first important and striptly
rigorous investigation of series was made by Gauss in con
nection with the hypergeometric series. The criterion devel
oped by him settles the question of convergence in every case
which it is intended to cover, and thus bears the stamp of
generality so characteristic of Gauss s writings. Owing to the
strangeness of treatment and unusual rigour, Gauss s paper
excited little interest among the mathematicians of that time.
More fortunate in reaching the public was Cauchy, whose
Analyse Algebrique of 1821 contains a rigorous treatment of
series. All series whose sum does not approach a fixed limit
as the number of terms increases indefinitely are called diver
gent. Like Gauss, he institutes comparisons with geometric
series, and finds that series with positive terms are convergent
or not, according as the wth root of the nth term, or the ratio
of the (n + l)th term and the nth term, is ultimately less
or greater than unity. To reach some of the cases where
these expressions become ultimately unity and fail, Cauchy
established two other tests. He showed that series with neg
ative terms converge when the absolute values of the terms
converge, and then deduces Leibniz s test for alternating
series. The product of two convergent series was not found
to be necessarily convergent. Cauchy s theorem that the
386 A HISTOBY OF MATHEMATICS,
product of two absolutely convergent series converges to the
product of the sums of the two series was shown half a cen
tury later by F. Mertens of G-raz to be still true if , of the two
convergent series to be multiplied together, only one is abso
lutely convergent.
The most outspoken critic of the old methods in series
was Abel. His letter to Ms friend Holmboe (1826) contains
severe criticisms. It is very interesting reading, even to
modern students. In his demonstration of the binomial theo
rem he established the theorem that if two series and their
product series are all convergent, then the product series will
converge towards the product of the sums of the two given
series. This remarkable result would dispose of the whole
problem of multiplication of series if we had a universal
practical criterion of convergency for semi-convergent series.
Since we do not possess such a criterion, theorems have been
recently established by A. Pringsheim of Munich and A. Voss
of Wiirzburg which remove in certain cases the necessity of
applying tests of convergency to the product series by the
application of tests to easier related expressions. Pringsheim
reaches the following interesting conclusions: The product,
of two semi-convergent series can never converge absolutely,
but a semi-convergent series, or even a divergent series, multi
plied by an absolutely convergent series, may yield an abso
lutely convergent product.
The researches of Abel and Cauchy caused a considerable
stir. We are told that after a scientific meeting in which
Cauchy had presented his first researches on series, Laplace
hastened home and remained there in seclusion until he had
examined the series in Ms M&oanique C&leste. Luckily, every
one was found to be convergent! We must not conclude,
however, that the new ideas at once displaced the old. On the
contrary, the new views were generally accepted only after a
ANALYSIS. 337
severe and long struggle. As late as 1844 De Morgan began a
paper on " divergent series " in this style : " I believe it will
be generally admitted that the heading of this paper describes
the only subject yet remaining, of an elementary character, on
which a serious schism exists among mathematicians as to the
absolute correctness or incorrectness of results,"
First in time in the evolution of more delicate criteria of
convergence and divergence come the researches of Josef Lud-
wig Eaabe (Crelle, Vol. IX.); then follow those of De Morgan
as given in his calculus. De Morgan established the loga
rithmic criteria which were discovered in part independently
by J. Bertrand. The forms of these criteria, as given by
Bertrand and by Ossian Bonnet, are more convenient than
De Morgan s. It appears from Abel s posthumous papers"
that he had anticipated the above-named writers in estab
lishing logarithmic criteria. It was the opinion of Bonnet
that the logarithmic criteria never fail ; but Du Bois-Rey-
mond and Pringsheim have each discovered series demon-
strably convergent in which these criteria fail to determine
the convergence. The criteria thus far alluded to have
been called by Fringsheim special criteria, because they all
depend upon a comparison of the nfh. term of the series
with special functions a n , n x , ^(logn)*, etc. Among the
first to suggest general criteria, and to consider the subject
from a still wider point of view, culminating in a regular
mathematical theory, was Kummer. He established a theorem
yielding a test consisting of two parts, the first part of
which was afterwards found to be superfluous. The study
of general criteria was continued by U. Dini of Pisa, Paul
Du Bois-Beyrnond, Gr. Kohn of Minden, and Pringsheim.
Du Bois-Reymond divides criteria into two classes : criteria
of the first kind and criteria of the second kind, according as
the general nth term, or the ratio of the (?i + l)th term and
338 A HISTORY OF MATHEMATICS,
the nth. term, is made the basis of research. Kummer s is a
criterion of the second kind. A criterion of the first kind,
analogous to this, was invented by Pringsheim. From the
general criteria established by Du Bois-Beyrnond and Prings
heim respectively, all the special criteria can be derived. The
theory of Pringsheim is very complete, and offers, in addition
to the criteria of the first kind and second kind, entirely new
criteria of a third Mud, and also generalised criteria of the
second kind, which apply, however, only to series with never
increasing terms. Those of the third kind rest mainly on the
consideration of the limit of the difference either of consecu
tive terms or of their reciprocals. In the generalised criteria
of the second kind he does not consider the ratio of two con
secutive terms, but the ratio of any two terms however far
apart, and deduces, among others, two criteria previously given
by Kohn and Ermakoff respectively.
Difficult questions arose in the study of Fourier s series. 79
Cauchy was the first who felt the necessity of inquiring into
its convergence. But his mode of proceeding was found
by Dirichlet to be unsatisfactory. Dirichlet made the first
thorough researches on this subject (Crelle, Vol. IV.). They
culminate in the result that whenever the function does not
become infinite, does not have an infinite number of dis
continuities, and does not possess an infinite number of
maxima and minima, then Fourier s series converges toward
the value of that function at all places, except points of
discontinuity, and there it converges toward the mean of
the two boundary values. Schlafli of Bern and Du Bois-
Eeymond expressed doubts as to the correctness of the mean
value, which were, however, not well founded. Dirichlet s
conditions are sufficient, but not necessary. Lipschitz, of
Bonn, proved that Fourier s series still represents the func
tion when the number of discontinuities is infinite, and
ANALYSIS. 339
established a condition on which, it represents a function
having an infinite number of maxima and minima. Dirich-
let s belief that all continuous functions can be represented
by Fourier s series at all points was shared by Eiemann and
H. Hankel, but was proved to be false by Du Bois-Keymond
and H. A. Schwarz.
Bdernann inquired what properties a function must have,
so that there may be a trigonometric series which, whenever
it is convergent, converges toward the value of the function.
He found necessary and sufficient conditions for this. They
do not decide, however, whether such a series actually repre
sents the function or not. Eiemann rejected Cauchy s defini
tion of a definite integral on account of its arbitrariness, gave
a new definition, and then inquired when a function has an
integral. His researches brought to light the fact that con
tinuous functions need not always have a differential coeffi
cient. But this property, which was shown by Weierstrass to
belong to large classes of functions, was not found necessarily
to exclude them from being represented by Fourier s series.
Doubts on some of the conclusions about Fourier s series were
thrown by the observation, made by Weierstrass, that the
integral of an infinite series can be shown to be equal to the
sum of the integrals of the separate terms only when the series
converges uniformly within the region in question. The sub
ject of uniform convergence was investigated by Philipp Lud-
wig Seidel (1848) and Gr. G-. Stokes (1847), and has assumed
great importance in Weierstrass theory of functions. It
became necessary to prove that a trigonometric series repre
senting a continuous function converges uniformly. This was
done by Heinrich Eduard Heine (1821-1881), of Halle. Later
researches on Fourier s series were made by G. Cantor and
Du Bois-Beymond.
As compared with the vast development of other mathe-
340 A HISTORY OF MATHEMATICS.
matical branclies ; the theory of probability has made very
insignificant progress since the time of Laplace. Improve
ments and sirnplications in the mode of exposition have been
made by A. De Morgan, G-. Boole, A. Meyer (edited by E.
Czuber), J. Bertrand. Cournot s and Westergaard s treatment
of insurance and the theory of life-tables are classical. Appli
cations of the calculus to statistics have been made by L. A. J.
Quetelet (1796-1874), director of the observatory at Brussels;
by Lexis ; Harald Westergaard, of Copenhagen ; and Dusing.
Worthy of note is the rejection of inverse probability by the
best authorities of our time. This branch of probability had
been worked out by Thomas Bayes (died 1761) and by Laplace
(Bk. II., Ch. VI. of his TMorie Analytique). By it some
logicians have explained induction. For example, if a man,
who has never heard of the tides, were to go to the shore of the
Atlantic Ocean and witness on m successive days the rise of the
sea, then, says Quetelet, he would be entitled to conclude that
there was a probability equal to that the sea would rise
next day. Putting m = 0, it is seen that this view rests upon
the unwarrantable assumption that the probability of a totally
unknown event is $, or that of all theories proposed for inves
tigation one-half are true. W. S. Jevons in his Principles of
Science founds induction upon the theory of inverse proba
bility, and F. Y, Edgeworth also accepts it in his Mathematical
Psychics.
The only noteworthy recent addition to probability is the
subject of " local probability," developed by several English
and a few American and French mathematicians. The earliest
problem on this subject dates back to the time of Buff on, the
naturalist, who proposed the problem, solved by himself and
Laplace, to determine the probability that a short needle,
thrown at random upon a floor ruled with equidistant parallel
ANALYSIS. 341
lines, will fall on one of the lines. Then came Sylvester s
four-point problem: to find the probability that four points,
taken at random within a given boundary, shall form a re
entrant quadrilateral. Local probability has been studied in
England by A. R. Clarke, H. McColl, S. Watson, J. Wolsten-
holme, but with greatest success by M. W. Crofton of the
military school at Woolwich. It was pursued in America by
E. B. Seitz ; in France by C. Jordan, E. Lemoine, E. Barbier,
and others. Through considerations of local probability,
Crofton was led to the evaluation of certain definite integrals.
The first full scientific treatment of differential equations
was given by Lagrange and Laplace. This remark is especially
true of partial differential equations. The latter were investi
gated in more recent time by Monge, Pfaff, Jacobi, iSmile Bour
(1831-1866) of Paris, A. Weiler, Clebsch, A. E". Korkine of St.
Petersburg, G-. Boole, A. Meyer, Cauchy, Serret, Sophus Lie,
and others. In 1873 their reseaches, on partial differential
equations of the first order, were presented in text-book form
by Paul Mansion, of the University of Gand. The keen
researches of Johann Friedrich Pfatf (1795-1825) marked a
decided advance. He was an intimate friend of young G-auss
at Gottingen. Afterwards he was with the astronomer Bode.
Later he became professor at Helmstadt, then at Halle. By a
peculiar method, Pfaff found the general integration of par
tial differential equations of the first order for any number
of variables. Starting from the theory of ordinary differential
equations of the first order in n variables, he gives first their
general integration, and then considers the integration of the
partial differential equations as a particular case of the former,
assuming, however, as known, the general integration of differ
ential equations of any order between two variables. His
researches led Jacobi to introduce the name "Pfaffian prob
lem." From the connection, observed by Hamilton, between
342 A HISTOKY OF MATHEMATICS.
a system of ordinary differential equations (in analytical
mechanics) and a partial differential equation, Jacobi drew
the conclusion that, of the series of systems whose successive
integration PfafPs method demanded, all but the first system
were entirely superfluous. Clebsch considered Pfaff s problem
from a new point of view, and reduced it to systems of simul
taneous linear partial differential equations, which can be
established independently of each other without any integra
tion. Jacobi materially advanced the theory of differential
equations of the first order. The problem to determine un
known functions in such a way that an integral containing
these functions and their differential coefficients, in a pre
scribed manner, shall reach a maximum or minimum value,
demands, in the first place, the vanishing of the first variation
of the integral. This condition leads to differential equations,
the integration of which determines the functions. To ascer
tain whether the value is a maximum or a minimum, the
second variation must be examined. This leads to new and
difficult differential equations, the integration of which, for
the simpler cases, was ingeniously deduced by Jacobi from
the integration of the differential equations of the first varia
tion. Jacobi s solution was perfected by Hesse, while Clebsch
extended to the general case Jacobi s results on the second
variation. Cauchy gave a method of solving partial differ
ential equations of the first order having any number of
variables, which was corrected and extended by Serret, J. Ber-
trand, 0. Bonnet in France, and Imschenetzky in Eussia.
Fundamental is the proposition of Cauchy that every ordinary
differential equation admits in the vicinity of any non-singular
point of an integral, which is synectic within a certain circle
of convergence, and is developable by Taylor s theorem.
Allied to the point of view indicated by this theorem is that
of Riemann, who regards a function of a single variable as
ANALYSIS. 343
defined by the position and nature of its singularities, and
who has applied this conception to that linear differential
equation of the second order, which is satisfied by the hyper-
geometric series. This equation was studied also by Gauss
and Kummer. Its general theory ; when no restriction is
imposed upon the yalue of the variable, has been considered
by J. Tannery ; of Paris, who employed Fuchs method of
linear differential equations and found all of Hummer s
twenty-four integrals of this equation. This study has been
continued by JMouard Goursat of Paris.
A standard text-book on Differential Equations, including
original matter on integrating factors, singular solutions, and
especially on symbolical methods, was prepared in 1859 by
George Boole (1815-1864), at one time professor in Queen s
University, Cork, Ireland. He was a native of Lincoln, and a
self-educated mathematician of great power. His treatise on
Finite Differences (1860) and his Laws of Thought (1854) are
works of high merit.
The fertility of the conceptions of Cauchy and Rlemann
with regard to differential equations is attested by the
researches to which they have given rise on the part of
Lazarus Fuchs of Berlin (born 1835), Felix Klein of G-ottingen
(born 1849), Henri Poincare of Paris (bom 1854), and others.
The study of linear differential equations entered a new
period with the publication of Fuchs memoirs of 1866
and 1868. Before this, linear equations with constant co
efficients were almost the only ones for which general methods
of integration were known. While the general theory of
these equations has recently been presented in a new light
by Herniite, Darboux, and Jordan, Fuchs began the study
from the more general standpoint of the linear differential
equations whose coefficients are not constant. He directed
his attention mainly to those whose integrals are all regular.
344 A HISTOBY OF MATHEMATICS.
If the variable be made to describe all possible paths enclos
ing one or more of the critical points of the equation, we
have a certain substitution corresponding to each of the
paths; the aggregate of all these substitutions being called
a group. The forms of integrals of such equations were
examined by Fuchs and by G-. Frobenius by independent
methods. Logarithms generally appear in the integrals of
a group, and Fuchs and Frobenius investigated the conditions
under which no logarithms shall appear. Through the study
of groups the reducibility or irreducibility of linear differ
ential equations has been examined by Frobenius and Leo
Konigsberger. The subject of linear differential equations,
not all of whose integrals are regular, has been attacked by
G. Frobenius of Berlin, W. Thome of G-reifswald (born 1841),
and Poinear6, but the resulting theory of irregular integrals
is as yet in very incomplete form.
The theory of invariants associated with linear differential
equations has been developed by Halphen and by A. B. Forsyth.
The researches above referred to are closely connected with
the theory of functions and of groups. Endeavours have thus
been made to determine the nature of the function defined by
a differential equation from the differential equation itself,
and not from any analytical expression of the function, obtained
first by solving the differential equation. Instead of studying
the properties of the integrals of a differential equation for all
the values of the variable, investigators at first contented them
selves with the study of the properties in the vicinity of a
given point. The nature of the integrals at singular points
and at ordinary points is entirely different. Albert Briot
(1817-1882) and Jem Claude Bouquet (1819-1885), both of
Paris, studied the case when, near a singular point, the dif
ferential equations take the form (a? a? ) = C(oy) . Fuchs
dx J
ANALYSIS. 345
gave the development in series of the integrals for the partic
ular case of linear equations. Poincare did the same for the
case when the equations are not linear, as also for partial
differential equations of the first order. The developments for
ordinary points were given by Cauchy and Madarae Kowalevsky.
The attempt to express the integrals hy developments that
are always convergent and not limited to particular points in
a plane necessitates the introduction of new transcendents, for
the old functions permit the integration of only a small num
ber of differential equations. Poincare* tried this plan with
linear equations, which were then the best known, having
been studied in the vicinity of given points by Euehs, Thome*,
Erobenius, Schwarz, Klein, and Halphen. Confining himself to
those with rational algebraical coefficients, Poincare" was able
to integrate them by the use of functions named by him Fuch-
sians. 81 He divided these equations into " families." If the
integral of such an equation be subjected to a certain trans
formation, the result will be the integral of an equation
belonging to the same family. The new transcendents have a
great analogy to elliptic functions ; while the region of the
latter may be divided . into parallelograms, each representing a
group, the former may be divided into curvilinear polygons,
so that the knowledge of the function inside of one polygon
carries with it the knowledge of it inside the others. Thus
Poincare" arrives at what he calls Fuchsian groups. He found,
moreover, that Euchsian functions can be expressed as the
ratio of two transcendents (theta-fuchsians) in the same way
that elliptic functions can be. If, instead of linear substitu
tions with real coefficients, as employed in the above groups,
imaginary coefficients be used, then discontinuous groups are
obtained, which he called Kleinians. The extension to non
linear equations of the method thus applied to linear equa
tions has been begun by Euchs and Poincare.
346 A HISTOEY OF MATHEMATICS.
We have seen that among the earliest of the several kinds
of " groups " are the finite discontinuous groups (groups in
the theory of substitution), which since the time of Galois
have become the leading concept in the theory of algebraic
equations; that since 1876 Felix Klein, H. Poincare, and
others have applied the theory of finite and infinite discontin
uous groups to the theory of functions and of differential
equations. The finite continuous groups were first made
the subject of general research in 1873 by Sophus Lie, now
of Leipzig, and applied by him to the integration of ordinary
linear partial differential equations.
Much interest attaches to the determination of those linear
differential equations which can be integrated by simpler
functions, such as algebraic, elliptic, or Abelian. This has
been studied by C. Jordan, P. Appel of Paris (born 1858), and
Poincare.
The mode of integration above referred to, which makes
known the properties of equations from the standpoint of the
theory of functions, does not suffice in the application of
differential equations to questions of mechanics. If we con
sider the function as defining a plane curve, then the general
form of the curve does not appear from the above mode of
investigation. It is, however, often desirable to construct
the curves defined by differential equations. Studies having
this end in view have been carried on by Briot and Bouquet,
and by Poincar& 81
The subject of singular solutions of differential equations
has been materially advanced since the time of Boole by G.
Darboux and Cayley. The papers prepared by these mathe
maticians point out a difficulty as yet unsurmounted : whereas
a singular solution, from the point of view of the integrated
equation, ought to be a phenomenon of universal, or at least of
general occurrence, it is, on the other hand, a very special and
THEORY OF FUNCTIONS. 847
exceptional phenomenon from the point of view of the differ
ential equation. 89 A geometrical theory of singular solutions
resembling the one used by Cayley was previously employed
by W. W. Johnson of Annapolis.
An advanced Treatise on Linear Differential Equations
(1889) was brought out by Thomas Craig of the Johns Hop
kins University. He chose the algebraic method of presenta
tion followed by Hermite and Poincare, instead of the geometric
method preferred by Klein and Schwarz. A notable work, the
Traite $ Analyse, is now being published by mile Picard of
Paris, the interest of which is made to centre in the subject of
differential equations.
THEORY OF FUNCTIONS.
We begin our sketch of the vast progress in the theory of
functions by considering the special class called elliptic func
tions. These were richly developed by Abel and Jacobi.
Mels Henrick Abel (1802-1829) was born at Findoe in Nor
way, and was prepared for the university at the cathedral
school in Christiania. He exhibited no interest in mathe
matics until 1818, when B. Holmboe became lecturer there,
and aroused Abel s interest by assigning original problems
to the class. Like Jacobi and many other young men who
became eminent mathematicians, Abel found the first exercise
of his talent in the attempt to solve by algebra the general
equation of the fifth degree. In 1821 he entered the Uni
versity in Christiania. The works of Euler, Lagrange, and
Legendre were closely studied by him. The idea of the inver
sion of elliptic functions dates back to this time. His extraor
dinary Success in mathematical study led to the offer of a
stipend by the government, that he might continue his studies
348 A HISTORY OF MATHEMATICS.
in Germany and Prance. Leaving Norway in 1825 ; Abel vis
ited the astronomer, Schumacher, in Hamburg, and spent six
months in Berlin, where he became intimate with August
Leopold Crelle (1780-1855), and met Steiner. Encouraged by
Abel and Steiner, Crelle started his journal in 1826. Abel
began to put some of his work in shape for print. His proof
of the impossibility of solving the general equation of the fifth
degree by radicals, first printed in 1824 in a very concise
form, and difficult of apprehension, was elaborated in greater
detail, and published in the first volume. He entered also
upon the subject of infinite series (particularly the binomial
theorem, of which he gave in Grelle s Journal a rigid general
investigation), the study of functions, and of the integral
calculus. The obscurities everywhere encountered by him
owing to the prevailing loose methods of analysis he endeav
oured to clear up. Tor a short time he left Berlin for Prei-
berg, where he had fewer interruptions to work, and it was
there that he made researches on hyperelliptic and Abelian
functions. In July, 1826, Abel left Germany for Paris with
out having met Gauss ! Abel had sent to Gauss his proof of
1824 of the impossibility of solving equations of the fifth
degree, to which Gauss never paid any attention. This slight,
and a haughtiness of spirit which he associated with Gauss,
prevented the genial Abel from going to Gottingen. A similar
feeling was entertained by him later against Cauchy. Abel
remained ten months in Paris. He met there Birichlet,
Legendre, Cauchy, and others ; but was little appreciated.
He had already published several important memoirs in
Crelle s Journal, but by the French this new periodical was
as yet hardly known to exist, and Abel was too modest to
speak of his own work. Pecuniary embarrassments induced
him to return home after a second short stay in Berlin. At
Christiania he for some time gave private lessons, and served
THEORY OF FUNCTIONS. 349
as decent. Crelle secured at last an appointment for Mm at
Berlin ; but the news of it did not reach Norway until after
the death of Abel at Proland. 82
At nearly the same time with Abel, Jacobi published articles
on elliptic functions. Legendre s favourite subject, so long
neglected, was at last to be enriched by some extraordinary
discoveries. The advantage to be derived by inverting the
elliptic integral of the first kind and treating it as a function
of its amplitude (now called elliptic function) was recognised
by Abel, and a few months later also by Jacobi. A second
fruitful idea, also arrived at independently by both, is the
introduction of imaginaries leading to the observation that
the new functions simulated at once trigonometric and expo
nential functions. For it was shown that while trigonometric
functions had only a real period, and exponential only an imag
inary, elliptic functions had both sorts of periods. These two
discoveries were the foundations upon which Abel and Jaeobi,
each in his own way, erected beautiful new structures. Abel
developed the curious expressions representing elliptic func
tions by infinite series or quotients of infinite products.
Great as were the achievements of Abel in elliptic functions,
they were eclipsed by his researches on what are now called
Abelian functions. Abel s theorem on these functions was
given by him in several forms, the most general of these
being that in his M6moire sur une propriety gen&rale d une
classe tr&s-6tendue de fonctions transcendentes (1826). The his
tory of this memoir is interesting. A few months after his
arrival in Paris, Abel submitted it to the French Academy.
Cauchy and Legendre were appointed to examine it ; but said
nothing about it until after Abel s death. In a brief statement
of the discoveries in question, published by Abel in Crette s
Journal, 1829, reference is made to that memoir. This led
Jacobi to inquire of Legendre whafr had become of it. Le-
350 A HISTOEY OF MATHEMATICS.
gendre says that the manuscript was so badly written as to be
illegible, and that Abel was asked to hand in a better copy,
which he neglected to do. The memoir remained in Canchy ; s
hands. It was not published until 1841. By a singular mis
hap, the manuscript was lost before the proof-sheets were
read.
In its form, the contents of the memoir belongs to the inte
gral calculus. Abelian integrals depend upon an irrational
function y which is connected with x by an algebraic equa
tion F(x } y) = 0. Abel s theorem asserts that a sum of such
integrals can be expressed by a definite number p of similar
integrals, where p depends merely on the properties of the
equation F(x, y) = 0. It was shown later that p is the defi
ciency of the curve F(x, y) = 0. The addition theorems of
elliptic integrals are deducible from Abel s theorem. The
hyperelliptic integrals introduced by Abel, and proved by him
to possess multiple periodicity, are special cases of Abelian
integrals whenever _p= or > 3. The reduction of Abelian to
elliptic integrals has been studied mainly by Jacobi, Hermite,
Konigsberger, Brioschi, Goursat, E. Picard, and 0. Bolza of
the University of Chicago.
Two editions of Abel s works have been published : the first
by Holmboe in 1839, and the second by Sylow and Lie in
1881.
Abel s theorem was pronounced by Jacobi the greatest dis
covery of our century on the integral calculus. The aged
Legendre, who greatly admired Abel s genius, called it "mon-
umentum aere perennius." During the few years of work
allotted to the young Norwegian, he penetrated new fields of
research, the development of which has kept mathematicians
busy for over half a century.
Some of the discoveries of Abel and Jacobi were anticipated
by Gauss. In the Disqidsitiones Arithmeticce he observed
THEOEY OF FUNCTIONS. 351
that the principles which he used in the division of the circle
were applicable to many other functions, besides the circular,
and particularly to the transcendents dependent on the integral
/dx
_^__. Erom this Jacobi 83 concluded that Gauss had
thirty years earlier considered the nature and properties of
elliptic functions and had discovered their double periodicity.
The papers in the collected works of Gauss confirm this con
clusion.
Carl Gustav Jacob Jacob! M (1804-1851) was born of Jewish
parents at Potsdam. Like many other mathematicians he was
initiated into mathematics by reading Euler. At the Univer
sity of Berlin, where he pursued his mathematical studies
independently of the lecture courses, he took the degree of
Ph.D. in 1825. After giving lectures in Berlin for two years,
he was elected extraordinary professor at Konigsberg, and two
years later to the ordinary professorship there. After the
publication of his Fimdamenta Nova he spent some time in
travel, meeting Gauss in Gottingen, and Legendre, Courier,
Poisson, in Paris. In 1842 he and his colleague, Bessel, at
tended the meetings of the British Association, where they
made the acquaintance of English mathematicians.
His early researches were on Gauss 7 approximation to the
value of definite integrals, partial differential equations, Le-
gendre s coefficients, and cubic residues. He read Legendre s
Exercises, which give an account of elliptic integrals. When
he returned the book to the library, he was depressed in spirits
and said that important books generally excited in him new
ideas, but that this time he had not been led to a single origi
nal thought. Though slow at first, his ideas flowed all the
richer afterwards. Many of his discoveries in elliptic func
tions were made independently by Abel. Jacobi communicated
his first researches to Crelle s Journal. In 1829, at the age
352 A HISTORY OP MATHEMATICS.
of twenty-five, he published Ms Fundaments Nova Theories
Functionum Ellipticarum, which contains in condensed form
the main results in elliptic functions. This work at once
secured for him a wide reputation. He then made a closer
study of theta-functions and lectured to his pupils on a new
theory of elliptic functions based on the theta-functions. He
developed a theory of transformation which led him to a mul
titude of formulae containing g, a transcendental function of
the modulus, defined by the equation q = e"** /*. He was also
led by it to consider the two new functions H and , which
taken each separately with two different arguments are the
four (single) theta-functions designated by the 1? 2 , 3; 4 . 56
In a short but very important memoir of 1S32, he shows that
for the hyperelliptic integral of any class the direct functions
to which Abel s theorem has reference are not functions of a
single variable, such as the elliptic sn, en, dn, but functions of
p variables. 56 Thus in the case p = 2, which Jacobi especially
considers, it is shown that Abel s theorem has reference to
two functions X(u, v), \i(u, v) } each of two variables, and
gives in effect an addition-theorem for the expression of the
functions X(u + u r , v + v ), X^u + u yV + v ) algebraically in
terms of the functions X(u, v), Xi(u, v), X(u r ,v ), Xi(u r ,v r ). By
the memoirs of Abel and Jacobi it may be considered that the
notion of the Abelian function of p variables was established
and the addition-theorem for these functions given. Eecent
studies touching Abelian functions have been made by Weier-
strass, E. Picard, Madame Kowalevski, and Poincare*. Jacobi s
work on differential equations, determinants, dynamics, and
the theory of numbers is mentioned elsewhere.
In 1842 Jacobi visited Italy for a few months to recuperate
his health. At this time the Prussian government gave Mm
a pension, and he moved to Berlin, where the last years of his
life were spent.
THEOBY OF FUNCTIONS. 353
The researches on functions mentioned thus far have been
greatly extended. In 1858 Charles Hennite of Paris (born 1822),
introduced in place of the variable q of Jacobi a new variable <o
connected with it by the equation q = e*, so that o> = ik /k, and
was led to consider the functions <(<o), ij/ (<*>), xW- 56 Henry
Smith regarded a theta-function with the argument equal to
zero, as a function of co. This he called an omega-function,
while the three functions <(o>), ^(w), xW? are n ^ s modular
functions. Researches on theta-functions with respect to real
and imaginary arguments have been made by Meissel of Kiel,
J. Thomae of Jena, Alfred Enneper of Gdttingen (1830-1885).
A general formula for the product of two theta-functions was
given in 1854 by H. Schroter of Breslau (1829-1892). These
functions have been studied also by Cauchy, Konigsberger of
Heidelberg (born 1837), E. S. Eichelot of Konigsberg (1808-
1875), Johann Georg Eosenhain of Konigsberg (1816-1887),
L. Schlani of Bern (bom 1818) *
Legendre s method of reducing an elliptic differential to its
normal form has called forth many investigations, most impor
tant of which are those of Eichelot and of Weierstrass of
Berlin.
The algebraic transformations of elliptic functions involve
a relation between the old modulus and the new one which
Jacobi expressed by a differential equation of the third order,
and also by an algebraic equation, called by him "modular
equation." The notion of modular equations was familiar to
Abel, but the development of this subject devolved upon later
investigators. These equations have become of importance in
the theory of algebraic equations, and have been studied by
Sohnke, E. Mathieu, L. Konigsberger, E. Betti of Pisa (died
1892), 0. Hermite of Paris, Joubert of Angers, Francesco
Brioschi of Milan. Schlani, H. Schroter, ML Gudermann of
Cleve, Gtitzlaff.
354 A HISTOEY OF MATHEMATICS.
Felix Klein of G-ottingen has made an extensive study of
modular functions, dealing with, a type of operations lying
between the two extreme types, known as the theory of substi
tutions and the theory of invariants and covariants. Klein s
theory las been presented in book-form by his pupil, Eobert
Fricke. The bolder features of it were first published in his
Ikosaeder, 1884. His researches embrace the theory of mod
ular functions as a specific class of elliptic functions, the
statement of a more general problem as based on the doctrine
of groups of operations, and the further development of the
subject in connection with a class of Bieniann s surfaces.
The elliptic functions were expressed by Abel as quotients
of doubly infinite products. He did not, however, inquire
rigorously into the convergency of the products. In 1845
Cayley studied these products, and found for them a complete
theory, based in part upon geometrical interpretation, which
he made the basis of the whole theory of elliptic functions.
Eisenstein discussed by purely analytical methods the general
doubly infinite product, and arrived at results which have
been greatly simplified in form by the theory of primary
factors, due to Weierstrass. A certain function involving a
doubly infinite product has been called by Weierstrass the
sigma-function, and is the basis of his beautiful theory of
elliptic functions. The first systematic presentation of Weier
strass theory of elliptic functions was published in 1886 by
G. H. Halphen in his TMorie des fonctions elliptiques et des
leurs applications. Applications of these functions have been
given also by A. G. Greenhill. Generalisations analogous to
those of Weierstrass on elliptic functions have been made
by Felix Klein on hyper elliptic functions.
Standard works on elliptic functions have been published by
Briot and Bouquet (1859), by Konigsberger, Cayley, Heinricli
Durtye of Prague (1821-1893), and others.
THEOBY OF FUNCTIONS. 355
Jacobins work on Abelian and theta-functions was greatly
extended by Adolpk Gopel (1812-1847), professor in a gym
nasium near Potsdam, and Johann Georg Rosenhain of Konigs-
berg (1816-1887). Gopel in Ms Theories transcendentium primi
ordmis admnbratio levis (Crelle, 35, 1847) and Rosenhain in
several memoirs established each independently, on the analogy
of the single theta-functions, the functions of two variables,
called double theta-functions, and worked out in connection
with them the theory of the Abelian functions of two variables.
The theta-relations established by G-opel and Bosenhain re
ceived for thirty years no further development, notwithstand
ing the fact that the double theta series came to be of increasing
importance in analytical, geometrical, and mechanical prob
lems, and that Hermite and Konigsberger had considered the
subject of transformation. Finally, the investigations of C. W.
Borchardt of Berlin (1817-1880), treating of the representation
of Kummer s surface by Gopel s biquadratic relation between
four theta-functions of two variables, and researches of H. H.
Weber of Marburg, F. Pryin of Wtirzburg, Adolf Krazer, and
Martin Krause of Dresden led to broader views. Eesearches
on double theta-functions, made by Cayley, were extended to
quadruple theta-functions by Thomas Craig of the Johns
Hopkins University.
Starting with the integrals of the most general form and
considering the inverse functions corresponding to these in
tegrals (the Abelian functions of p variables), Eiemann
defined the theta-functions of p variables as the sum of a
p-tuply infinite series of exponentials, the general term de
pending on p variables. Hiemann shows that the Abelian
functions are algebraically connected with theta-functions of
the proper arguments, and presents the theory in the broadest
form. 56 He rests the theory of the multiple theta-functions
upon the general principles of the theory of functions of a
complex variable.
856 A HISTORY OF MATHEMATICS.
Through the researches of A. Brill of Tubingen, M. Nother
of Erlangen, and Ferdinand Lindemann of Munich, made
in connection with Biemann-Roch s theorem and the theory
of residuation, there has grown out of the theory of Abelian
functions a theory of algebraic functions and point-groups on
algebraic curves.
Before proceeding to the general theory of functions, we
make mention of the " calculus of functions," studied chiefly
by C. Babbage, J. F. W. Herschel, and De Morgan, which was
not so much a theory of functions as a theory of the solution
of functional equations by means of known functions or
symbols.
The history of the general theory of functions begins with
the adoption of new definitions of a function. With the
Bernoullis and Leibniz, y was called a function of #, if there
existed an equation between these variables which made it
possible to calculate y for any given value of x lying any
where between oo and + oo. The study of Fourier s theory
of heat led Dirichlet to a new definition : y is called a function
of x, if y possess one or more definite values for each of certain
values that x is assumed to take in an interval x to %. In
functions thus defined, there need be no -analytical connection
between y and x, and it becomes necessary to look for possible
discontinuities. A great revolution in the ideas of a function
was brought about by Cauchy when, in a function as defined
by Dirichlet, he gave the variables imaginary values, and when
he extended the notion of a definite integral by letting the
variable pass from one limit to the other by a succession of
imaginary values along arbitrary paths. Cauchy established
several fundamental theorems, and gave the first great impulse
to the study of the general theory of functions. His researches
were continued in France by Puiseux and Liouville. But more
profound investigations were made in Germany by Blemann.
THEORY OF FUNCTIONS. 357
Georg Friedricli Berahard Riemann (1826-1866) was born at
Breselenz in Hanover. His father wished him to study
theology, and he accordingly entered upon philological and
theological studies at Gottingen. He attended also some lec
tures on mathematics. Such was his predilection for this
science that he abandoned theology. After studying for a
time under G-auss and Stern, he was drawn, in 1847, to Berlin
by a galaxy of mathematicians, in which shone Dirichlet,
Jacobi, Steiner, and Eisenstein. Returning to Gottingen in
1850, he studied physics under Weber, and obtained the
doctorate the following year. The thesis presented on that
occasion, Grundlagen fur eine allgemeine Theorie der Funktionen
einer verdnderlichen complexen Grosse, excited the admiration of
Gauss to a very unusual degree, as did also Biemann s trial
lecture, Ueber die Hypotliesen welche der Geometrie zu Grunde
liegen. Biemann s Habilitationsschrift was on the Bepresen-
tation of a Function by means of a Trigonometric Series, in
which he advanced materially beyond the position of Dirich
let. Our hearts are drawn to this extraordinarily gifted but
shy genius when we read of the timidity and nervousness
displayed when he began to lecture at Gottingen, and of his
jubilation over the unexpectedly large audience of eight students
at his first lecture on differential equations.
Later he lectured on Abelian functions to a class of three
only, Schering, Bjerknes, and Dedekind. Gauss died in 1855,
and was succeeded by Dirichlet. On the death of the latter,
in 1859, B/iemann was made ordinary professor. In 1860 he
visited Paris, where he made the acquaintance of French
mathematicians. The delicate state of his health induced
him to go to Italy three times. He died on his last trip at
Selasca, and was buried at Biganzolo.
Like all of Biemann s researches, those on functions were
profound and far-reaching. He laid the foundation for a
358 A HISTORY OF MATHEMATICS,
general theory of functions of a complex variable. The theory
of potential, which up to that time had been used only in
mathematical physics, was applied by him in pure mathe
matics. He accordingly based his theory of functions on the
f\2 f\2
partial differential equation, ^ -j- -^ = Aw = 0, which must
r dx 2 dy 2
hold for the analytical function w = u + iv of z = x + iy. It
had been proved by Dirichlet that (for a plane) there is always
one, and only one, function of x and y, which satisfies Aw = 0,
and which, together with its differential quotients of the first
two orders, is for all values of x and y within a given area
one-valued and continuous, and which has for points on the
boundary of the area arbitrarily given values. 86 Eiemann
called this " Dirichlet s principle," but the same theorem was
stated by Green and proved analytically by Sir William
Thomson. It follows then that w is uniquely determined for
all points within a closed surface, if u is arbitrarily given
for all points on the curve, whilst v is given for one point
within the curve. In order to treat the more complicated
case where w has n values for one value of z, and to observe
the conditions about continuity, Eiemann invented the cele
brated surfaces, known as "Eiemann s surfaces," consisting
of n coincident planes or sheets, such that the passage from
one sheet to another is made at the branch-points, and that the
n sheets form together a multiply-connected surface, which
can be dissected by cross-cuts into a singly-connected surface.
The 7i-valued function w becomes thus a one-valued function.
Aided by researches of J. Liiroth of Ereiburg and of Clebsch,
W. 3L Clifford brought Eiemann s surface for algebraic functions
to a canonical form, in which only the two last of the n leaves
are multiply-connected, and then transformed the surface into
the surface of a solid with p holes. A. Hurwitz of Zurich
discussed the question, how far a Eiemann s surface is deter-
THEOBY OF FUNCTIONS. 359
ruinate by the assignment of its number of sheets, its branch
points and branch-lines. 62
Eiemann s theory ascertains the criteria which will deter
mine an analytical function by aid of its discontinuities and
boundary conditions, and thus defines a function indepen
dently of a mathematical expression. In order to show that
two different expressions are identical, it is not necessary to
transform one into the other, but it is sufficient to prove the
agreement to a far less extent, merely in certain critical points.
Eiemann s theory, as based on Dirichlet s principle (Thom
son s theorem), is not free from objections. It has become
evident that the existence of a derived function is not a con
sequence of continuity, and that a function may be integrable
without being differentiable. It is not known how far the
methods of the infinitesimal calculus and the calculus of
variations (by which Dirichlet s principle is established) can
be applied to an unknown analytical function in its generality.
Hence the use of these methods will endow the functions with
properties which themselves require proof. Objections of this
kind to Eiemann s theory have been raised by Kronecker,
Weierstrass, and others, and it has become doubtful whether
his most important theorems are actually proved. In con
sequence of this, attempts have been made to graft Eiemann s
speculations on the more strongly rooted methods of Weier
strass. The latter developed a theory of functions by start
ing, not with the theory of potential, but with analytical
expressions and operations. Both applied their theories to
Abelian functions, but there Eiemann s work is more gen
eral. 86
The theory of functions of one complex variable has been
studied since Eiemann s time mainly by Karl Weierstrass of
Berlin (born 1815), Gustaf Mittag-Leffler of Stockholm (born
1846), and Poincare of Paris. Of the three classes of such
860 A HISTOBY OF MATHEMATICS.
functions (viz. functions uniform throughout, functions uni
form only in lacunary spaces, and non-uniform functions)
Weierstrass showed that those functions of the first class
which can he developed according to ascending powers of x
into converging series, can be decomposed into a product of an
infinite number of primary factors. A primary factor of the
species n is the product (1 j e p ( x ), P (x y being an entire poly-
V a J
nomial of the wfch degree. A function of the species n is one,
all the primary factors of which are of species n. This classi
fication gave rise to many interesting problems studied also
by Poincare.
The first of the three classes of functions of a complex
variable embraces, among others, functions having an infinite
number of singular points, but no singular lines, and at the
same time no isolated singular points. These are Fuchsian
functions, existing throughout the whole extent. Poincare
first gave an example of such a function.
Uniform functions of two variables, unaltered by certain
linear substitutions, called hyperfuchsian functions, have been
studied by E. Picard of Paris, and by Poincare. 81
Functions of the second class, uniform only in lacunary
spaces, were first pointed out by Weierstrass. The Fuchsian
and the Kleinian functions do not generally exist, except in
the interior of a circle or of a domain otherwise bounded, and
are therefore examples of functions of the second class.
Poincar has shown how to generate functions of this class,
and has studied them along the lines marked out by Weier
strass. Important is his proof that there is no way of
generalising them so as to get rid of the lacunae.
Non-uniform functions are much less developed than the
preceding classes, even though their properties in the vicinity
of a given point have been diligently studied, and though
THEORY OF FUNCTIONS. 361
much, light has been thrown on them by the use of Bdemann s
surfaces. With the view of reducing their study to thai of
uniform transcendents, Poincare proved that if y is any
analytical non-uniform function of x, one can always find a
variable z, such that x and y are uniform functions of z.
Weierstrass and Darboux have each given examples of con
tinuous functions having no derivatives. Formerly it had
been generally assumed that every function had a derivative.
Ampere was the first who attempted to prove analytically
(1806) the existence of a derivative, but the demonstration
is not valid. In treating of discontinuous functions, Darboux
established rigorously the necessary and sufficient condition
that a continuous or discontinuous function be susceptible of
integration. He gave fresh evidence of the care that must
be exercised in the use of series by giving an example of a
series always convergent and continuous, such that the series
formed by the integrals of the terms is always convergent, and
yet does not represent the integral of the first series. 87
The general theory of functions of two variables has been
investigated to some extent by Weierstrass and Poincare.
H. A. Schwarz of Berlin (born 1845), a pupil of Weierstrass,
has given the conform representation (Abbildung) of various
surfaces on a circle. In transforming by aid of certain
substitutions a polygon bounded by circular arcs into another
also bounded by circular arcs, he was led to a remarkable
differential equation \f;(u , f) = \l/(u, t), where $ (u, t) is the
expression which Cayley calls the "Schwarzian derivative,"
and which led Sylvester to the theory of reciprocants.
Schwarz s developments on minimum surfaces, his work on
hypergeometric series, his inquiries on the existence of solu
tions to important partial differential equations under prescribed
conditions, have secured a prominent place in mathematical
literature.
362 A HISTOEY OF MATHEMATICS.
The modern theory of functions of one real variable was
first worked out by H. Hankel, Dedekind, G. Cantor, Dini, and
Heine, and then carried further, principally, by Weierstrass,
Schwarz, Du Bois-Reymond, Thoniae, and Darboux. Hankel
established the principle of the condensation of singularities ;
Dedekind and Cantor gave definitions for irrational numbers ;
definite integrals were studied by Thomae, Du Bois-Reymond,
and Darboux along the lines indicated by the definitions of
such integrals given by Cauchy, Dirichlet, and Eiemann. Dini
wrote a text-book on functions of a real variable (1873), which
was translated into German, with additions, by J. Liiroth and
A. Schepp. Important works on the theory of functions are
the GOUTS de M. Hermite; Tannery s TMorie des Fonctions
d une variable seule, A Treatise on the Theory of Functions by
James Harlcness and Frank Morley, and Theory of Functions of
a Complex Variable by A. R. Forsytli.
THEORY OF NUMBERS.
"Mathematics, the queen of the sciences, and arithmetic,
the queen of mathematics." Such was the dictum of Gauss,
who was destined fc. olutionise the theory of numbers.
When asked who was the greatest mathematician in Ger
many, Laplace answered, PfafF. When the questioner said
he should have thought Gauss was, Laplace replied, "Pfaff
is by far the giuatest mathematician in Germany ; but Gauss
is the greatest in all Europe." 83 Gauss is one of the three
greatest masters of modern analysis, Lagrange, Laplace,
Gauss. Of these three contemporaries he was the youngest.
While the first two belong to the period in mathematical his
tory preceding the one now under consideration, Gauss is the
one whose writings may truly be said to mark the beginning
THEORY OF NUMBERS. 363
of our own epoch. In Mm that abundant fertility of inven
tion, displayed by mathematicians of the preceding period,
is combined with an absolute rigorousness in demonstration
which is too often wanting in their writings, and which the
ancient Greeks might have envied. Unlike Laplace, Gauss
strove in his writings after perfection of form. He rivals
Lagrange in elegance, and surpasses this great Frenchman in
rigour. "Wonderful was his richness of ideas 5 one thought fol
lowed another so quickly that he had hardly time to write
down even the most meagre outline. At the age of twenty
Gauss had overturned old theories and old methods in all
branches of higher mathematics ; but little pains did he take
to publish his results, and thereby to establish his priority.
He was the first to observe rigour in the treatment of infinite
series, the first to fully recognise and emphasise the impor
tance, and to make systematic use of determinants and of
imaginaries, the first to arrive at the method of least squares,
the first to observe the double periodicity of elliptic functions.
He invented the heliotrope and, together with Weber, the
bifilar magnetometer and the declination instrument. He
reconstructed the whole of magnetic science.
Carl Friedricli Gauss 47 (1777-1855), the son, of a bricklayer,
was born at Brunswick. He used to say, jokingly, that he
could reckon before he could talk. The marvellous aptitude
for calculation of the young boy attracted the attention of
Bartels, afterwards professor of mathematics at Dorpat, who
brought him under the notice of Charles "William, Duke of
Brunswick. The duke undertook to educate the boy, and sent
him to the Collegium Carolinum. His progress in languages
there was quite equal to that in mathematics. In 1795 he
went. to Gottingen, as yet undecided whether to pursue philol
ogy or mathematics. Abraham Gotthelf Kastner, then pro
fessor of mathematics there, and now chiefly remembered for
364 A HISTORY OF MATHEMATICS.
his Geschichte der Mathematik (1796), was not an inspiring
teacher. At the age of nineteen Gauss discovered a method of
inscribing in a circle a regular polygon of seventeen sides, and
this success encouraged him to pursue mathematics. He worked
quite independently of his teachers, and while a student at
Gottingen made several of his greatest discoveries. Higher
arithmetic was his favourite study. Among his small circle
of intimate friends was Wolfgang Bolyai. After completing
his course he returned to Brunswick. In 1798 and 1799 he
repaired to the university at Helmstadt to consult the library,
and there made the acquaintance of Pfaff, a mathematician of
much power. In 1807 the Emperor of Eussia offered Gauss a
chair in the Academy at St. Petersburg, but by the advice of
the astronomer Olbers, who desired to secure him as director
of a proposed new observatory at Gottingeni he declined the
offer, and accepted the place at Gottingen. Gauss had a
marked objection to a mathematical chair, and preferred the
post of astronomer, that he might give all his time to science.
He spent his life in Gottingen in the midst of continuous
work. In 1828 he went to Berlin to attend a meeting of
scientists, but after this he never again left Gottingen, except
in 1854, when a railroad was opened between Gottingen and
Hanover. He had a strong will, and his character showed
a curious mixture of self-conscious dignity and child-like
simplicity. He was little communicative, and at times
morose.
A new epoch in the theory of numbers dates from the publi
cation of his Disquisitiones Arithmeticce, Leipzig, 1801. The
beginning of this work dates back as far as 1795. Some of its
results had been previously given by Lagrange and Euler, but
were reached independently by Gauss, who had gone deeply
into the subject before he became acquainted with the writ
ings of his great predecessors. The Disquisitiones Arithmetics
THEORY OF NUMBERS. 365
was already in print when Legendre s Theorie des Nombres
appeared. The great law of quadratic reciprocity, given in
the fourth section of Gauss work, a law which involves the
whole theory of quadratic residues, was discovered "by him by
induction before he was eighteen, and was proved by him one
year later. Afterwards he learned that Euler had imperfectly
enunciated that theorem, and that Legendre had attempted
to prove it, but met with apparently insuperable difficulties.
In the fifth section Gauss gave a second proof of this " gem "
of higher arithmetic. In 1808 followed a third and fourth
demonstration; in 1817, a fifth and sixth. ISTo wonder that
he felt a personal attachment to this theorem. Proofs were
given also by Jacobi, Eisenstein, Liouville, Lebesgue, A.
Genocchi, Kummer, M. A. Stern, Chr. Zeller, Kronecker,
Bouniakowsky, E. Schering, J. Petersen, Voigt, E. Busche,
and Th. Pepin. 48 The solution of the problem of the repre
sentation of numbers by binary quadratic forms is one of the
great achievements of Gauss. He created a new algorithm by
introducing the theory of congruences. The fourth section
of the Disquisitiones AritJimeticce, treating of congruences of
the second degree, and the fifth section, treating of quadratic
forms, were, until the time of Jacobi, passed over with universal
neglect, but they have since been the starting-point of a long
series of important researches. The seventh or last section,
developing the theory of the division of the circle, was received
from the start with deserved enthusiasm, and has since been
repeatedly elaborated for students. A standard work on
Kreistheilung was published in 1872 by Paul Bachmann, then
of Breslau. Gauss had planned an eighth section, which was
omitted to lessen the expense of publication. His papers on
the theory of numbers were not all included in his great treatise.
Some of them were published for the first time after his death
in his collected works (1863-1871). He wrote two memoirs on
366 A HISTORY OF MATHEMATICS.
the theory of biquadratic residues (1825 and 1831), the second
of which contains a theorem of biquadratic reciprocity.
Gauss was led to astronomy by the discovery of the planet
Ceres at Palermo in 1801. His determination of the elements
of its orbit with sufficient accuracy to enable Olbers to redis
cover it, made the name of Gauss generally known. In 1809
he published the Theoria motus corporum coelestium, which
contains a discussion of the problems arising in the deter
mination of the movements of planets and comets from
observations made on them under any circumstances. In it
are found four formulae in spherical trigonometry, now usually
called "Gauss Analogies," but which were published some
what earlier by Karl Brandon Mollweide of Leipzig (1774-
1825), and earlier still by Jean Baptiste Joseph Delambre
(1749-1822) . M Many years of hard work were spent in the
astronomical and magnetic observatory. He founded the
German Magnetic Union, with the object of securing con
tinuous observations at fixed times. He took part in geodetic
observations, and in 1843 and 1846 wrote two memoirs, Ueber
Gegenstdnde der hoheren Geodesie. He wrote on the attrac
tion of homogeneous ellipsoids, 1813. In a memoir on capil
lary attraction, 1833, he solves a problem in the calculus of
variations involving the variation of a certain double integral,
the limits of integration being also variable ; it is the earliest
example of the solution of such a problem. He discussed the
problem of rays of light passing through a system of lenses.
Among Gauss pupils were Christian Heinrich Schumacher,
Christian Gerling, Friedrich Mcolai, August Ferdinand
Mobius, Georg Wilhelm Struve, Johann Frantz Encke.
Gauss researches on the theory of numbers were the start
ing-point for a school of writers, among the earliest of whom
was Jacobi. The latter contributed to Crelle s Journal an article
on cubic residues, giving theorems without proofs. After the
THEOBY OF NUMBEES. 367
publication of Gauss 3 paper on biquadratic residues, giving
the law of biquadratic reciprocity, and his treatment of com
plex numbers, Jacobi found a similar law for cubic residues.
By the theory of elliptical functions, he was led to beautiful
theorems on the representation of numbers by 2, 4, 6, and 8
squares. Next come the researches of Dirichlet, the expounder
of Gauss, and a contributor of rich results of his own.
Peter Gustav Lejeune Dirichlet 88 (1805-1859) was born in
Duren, attended the gymnasium in Bonn, and then the
Jesuit gymnasium in Cologne. In 1822 he was attracted to
Paris by the names of Laplace, Legendre, Fourier, Poisson,
Cauchy. The facilities for a mathematical education there
were far better than in Germany, where Gauss was the only
great figure. He read in Paris Gauss Disquisitiones Arith-
meticce, a work which he never ceased to admire and study.
Much in it was simplified by Dirichlet, and thereby placed
within easier reach of mathematicians. His first memoir on
the impossibility of certain indeterminate equations of the
fifth degree was presented to the French Academy in 1825.
He showed that Fermat s equation, x n + y n = z n , cannot exist
when ,<ft = 5. Some parts of the analysis are, however,
Legendre s. Euler and Lagrange had proved this when n is 3
and 4, and Lame proved it when n = 7. Dirichlet s acquaint
ance with Fourier led him to investigate Fourier s series. He
became decent in Breslau in 1827. In 1828 he accepted a
position in Berlin, and finally succeeded Gauss at Gottingen
in 1855. The general principles on which depends the aver
age number of classes of binary quadratic forms of positive
and negative determinant (a subject first investigated by
Gauss) were given by Dirichlet in a memoir, Ueber die Bestim-
mung der mittleren Werthe in der Zahlentheorie, 1849. More
recently F. Mertens of Graz has determined the asymptotic
values of several numerical functions. Dirichlet gave some
368 A HISTORY OF MATHEMATICS.
attention to prime numbers. G-auss and Legendre had given
expressions denoting approximately the asymptotic value of
the number of primes inferior to a given limit, but it remained
for Bdemann in his memoir, Ueber die Anzakl der Primzahlen
unter einer gegebenen Gfrosse, 1859, to give an investigation of
the asymptotic frequency of primes which is rigorous. Ap
proaching the problem from a different direction, Patnutij
Tchebyclieffy formerly professor in the University of St. Peters
burg (born 1821), established, in a celebrated memoir, Sur les
Nombres Premiers, 1850, the existence of limits within which
the sum of the logarithms of the primes P, inferior to a given
number a?, must be comprised. 89 This paper depends on very
elementary considerations, and, in that respect, contrasts
strongly with Riemann s, which involves abstruse theorems
of the integral calculus. Poincare s papers, Sylvester s con
traction of Tchebycheff s limits, with reference to the distri
bution of primes, and researches of J. Hadamard (awarded the
Grand prix of 1892), are among the latest researches in this
line. The enumeration of prime numbers has been undertaken
at different times by various mathematicians. In 1877 the
British Association began, the preparation of factor-tables,
under the direction of J. W. L. G-laisher. The printing, by
the Association, of tables for the sixth million marked the
completion of tables, to the preparation of which Germany,
France, and England contributed, and which enable us to
resolve into prime factors every composite number less than
9,000,000.
Miscellaneous contributions to the theory of numbers were
made by Cauchy. He showed, for instance, how to find all
the infinite solutions of a homogeneous indeterminate equation
of the second degree in three variables when one solution is
given. He established the theorem that if two congruences,
which have the same modulus, admit of a common solution,
THEORY OF NUMBERS. 369
the modulus is a divisor of their resultant. Joseph Liouville
(1809-1882), professor at the College de France, investigated
mainly questions on the theory of quadratic forms of two, and
of a greater number of variables. Profound researches were
instituted by Ferdinand Gotthold Eisenstein (1823-1852), of
Berlin. Ternary quadratic forms had been studied somewhat
by Gauss, but the extension from two to three indeterminates
was the work of Eisenstein who, in his memoir. Neue Tkeo-
reme der liolieren Arithmetik, defined the ordinal and generic
characters of ternary quadratic forms of uneven determinant;
and, in case of definite forms, assigned the weight of any order
or genus. But he did not publish demonstrations of his re
sults. In inspecting the theory of binary cubic forms, he was
led to the discovery of the first covariant ever considered in
analysis. He showed that the series of theorems, relating to
the presentation of numbers by sums of squares, ceases when
the number of squares surpasses eight. Many of the proofs
omitted by Eisenstein were supplied by Henry Smith, who
was one of the few Englishmen who devoted themselves to the
study of higher arithmetic.
Henry John Stephen Smith 90 (1826-1883) was born in Lon
don, and educated at Eugby and at Balliol College, Oxford.
Before 1847 he travelled much in Europe for his health,
and at one time attended lectures of Arago in Paris, but
after that year he was never absent from Oxford for a
single term. In 1861 he was elected Savilian professor of
geometry. His first paper on the theory of numbers appeared
in 1855. The results of ten years study of everything pub
lished on the theory of numbers are contained in his Eeports
which appeared in the British Association volumes from 1859
to 1865. These reports are a model of clear and precise
exposition and perfection of form. They contain much orig
inal matter, but the chief results of his own discoveries were
870 A HISTOBY OF MATHEMATICS.
printed in the Philosophical Transactions for 1861 and 1867.
They treat of linear indeterminate equations and congruences,
and of the orders and genera of ternary quadratic forms. He
established the principles on which the extension to the gen
eral case of n indeterminates of quadratic forms depends.
He contributed also two memoirs to the Proceedings of the
Royal Society of 1864 and 1868, in the second of which he
remarks that the theorems of Jacobi, Eisenstein, and Liou-
ville, relating to the representation of numbers by 4, 6, 8
squares, and other simple quadratic forms are dedueible by a
uniform method from the principles indicated in his paper.
Theorems relating to the case of 5 squares were given by
Eisenstein, but Smith completed the enunciation of them, and
added the corresponding theorems for 7 squares. The solu
tion of the cases of 2, 4, 6 squares may be obtained by elliptic
functions, but when the number of squares is odd, it involves
processes peculiar to the theory of numbers. This class of
theorems is limited to 8 squares, and Smith completed the
group. In ignorance of Smith s investigations, the Erench
Academy offered a prize for the demonstration and comple
tion of Eisenstein s theorems for 5 squares. This Smith had
accomplished fifteen years earlier. He sent in a dissertation
in 1882, and next year, a month after his death, the prize was
awarded to him, another prize being also awarded to H. Min-
kowsky of Bonn. The theory of numbers led Smith to the
study of elliptic functions. He wrote also on modern geome
try. His successor at Oxford was J. J. Sylvester.
Ernst Eduard Kummer (1810-1893), professor in the Uni
versity of Berlin, is closely identified with the theory of num
bers. Dirichlet s work on complex numbers of the form a-M&,
introduced by Gauss, was extended by him, by Eisenstein,
and Dedekind. Instead of the equation x* 1 = 0, the roots
of which yield Gauss units, Eisenstein used the equation
THEORY OF NUMBERS. 371
o 3 1 = and complex numbers a + 6/> (p being a cube root
of unity), the theory of which resembles that of Gauss num
bers. Kummer passed to the general case x n 1 = and got
complex numbers of the form a = a 1 A 1 + a 2 A 2 + a s A s -\ ,
where a t are whole real numbers, and A+ roots of the above
equation. 59 Euclid s theory of the greatest common divisor
is not applicable to such complex numbers, and their prime
factors cannot be defined in the same way as prime factors of
common integers are denned. In the effort to overcome this
difficulty, Kummer was led to introduce the conception of
"ideal numbers." These ideal numbers have been applied by
Gr. Zolotareff of St. Petersburg to the solution of a problem
of the integral calculus, left unfinished by Abel (LiouviHe s
Journal, Second Series, 1864, Vol. IX.). Julius Wttlielm Richard
Dedekind of Braunschweig (born 1831) has given in the second
edition of Dirichlet s Vorlesungen uber ZahlentJieorie a new
theory of complex numbers, in which he to some extent
deviates from the course of Kummer, and avoids the use of
ideal numbers. Dedekind has taken the roots of any irreduci
ble equation with integral coefficients as the units for his com
plex numbers. Attracted by Kummer s investigations, his
pupil, Leopold Kronecker (1823-1891) made researches which
he applied to algebraic equations.
On the other hand, efforts have been made to utilise in the
theory of numbers the results of the modern higher algebra.
Following up researches of Hermite, Paul Bachmann of Munster
investigated the arithmetical formula which gives the auto-
morphics of a ternary quadratic form. 89 The problem -of the
equivalence of two positive or definite ternary quadratic forms
was solved by L. Seeber ; and that of the arithmetical auto-
morphics of such forms, by Eisenstein. The more difficult prob
lem of the equivalence for indefinite ternary forms has been
investigated by Edward Selling of Wtirzburg. On quadratic
372 A HISTOBY OF MATHEMATICS.
forms of four or more indeterminates little lias yet been done.
Hermite snowed that the number of non-equivalent classes of
quadratic forms having integral coefficients and a given dis
criminant is finite, while Zolotareif and A. HT. Korkine, both
of St. Petersburg, investigated the minima of positive quadratic
forms. In connection with binary quadratic forms, Smith
established the theorem that if the joint invariant of two
properly primitive forms vanishes, the determinant of either
of them is represented primitively by the duplicate of the
other.
The interchange of theorems between arithmetic and algebra
is displayed in the recent researches of J. W. L. G-laisher
of Trinity College (born 1848) and Sylvester. Sylvester gave
a Constructive Theory of Partitions, which received additions
from his pupils, If. Eranklin and G. S. Ely.
The conception of " number" has been much extended in
our time. With the (keeks it included only the ordinary
positive whole numbers ; Diophantus added rational fractions
to the domain of numbers. Later negative numbers and
imaginaries came gradually to be recognised. Descartes fully
grasped the notion of the negative ; Gauss, that of the imagi
nary. With Euclid, a ratio, whether rational or irrational, was
not a number. The recognition of ratios and irrationals as
numbers took place in the sixteenth century, and found expres
sion with Kewton. By the ratio method, the continuity of the
real number system has been based on the continuity of space,
but in recent time three theories of irrationals have been
advanced by Weierstrass, J. W. B. Dedekind, G. Cantor, and
Heine, which prove the continuity of numbers without borrow
ing it from space. They are based on the definition of numbers
by regular sequences, the use of series and limits, and some
new mathematical conceptions.
APPLIED MATHEMATICS. 373
APPLIED MATHEMATICS.
Notwithstanding the beautiful developments of celestial
mechanics reached by Laplace at the close of the eighteenth
century, there was made a discovery on the first day of the
present century which presented a problem seemingly beyond
the power of that analysis. We refer to the discovery of Ceres
by Piazzi in Italy, which became known in Germany just after
the philosopher Hegel had published a dissertation proving a
priori that such a discovery could not be made. From the
positions of the planet observed by Piazzi its orbit could not
be satisfactorily calculated by the old methods, and it remained
for the genius of G-auss to devise a method of calculating
elliptic orbits which was free from the assumption of a small
eccentricity and inclination. Gauss method was developed
further in his Theoria Motus. The new planet was re-dis
covered with aid of Gauss data by Olbers, an astronomer
who promoted science not only by Ms own astronomical
studies, but also by discerning and directing towards astro
nomical pursuits the genius of Bess el.
Friedrich Wilhelm Bessel 91 (1784-1846) was a native of
Minden in Westphalia. Fondness for figures, and a distaste
for Latin grammar led him to the choice of a mercantile
career. In his fifteenth year he became an apprenticed clerk
in Bremen, and for nearly seven years he devoted his days to
mastering the details of his business, and part of his nights to
study. Hoping some day to become a supercargo on trading
expeditions, he became interested in observations at sea. With
a sextant constructed by him and an ordinary clock he deter
mined the latitude of Bremen. His success in this inspired
him for astronomical study. One work after another was
mastered by him, unaided, during the hours snatched from
374 A HISTORY OF MATHEMATICS.
sleep. From old observations he calculated the orbit of
Halley s comet. Bessel introduced himself to Gibers, and
submitted to him the calculation, which Olbers immediately
sent for publication. Encouraged by Olbers, Bessel turned
his back to the prospect of affluence, chose poverty and the
stars, and became assistant in J. H. Schroter s observatory at
Lilienthal. Four years later he was chosen to superintend
the construction of the new observatory at Konigsberg. 92 la
the absence of an adequate mathematical teaching force, Bessel
was obliged to lecture on mathematics to prepare students for
astronomy. He was relieved of this work in 1825 by the
arrival of Jacobi. We shall not recount the labours by which
Bessel earned the title of founder of modern practical astron
omy and geodesy. As an observer he towered far above
G-auss, but as a mathematician he reverently bowed before the
genius of his great contemporary. Of BessePs papers, the one
of greatest mathematical interest is an " UntersucJiung des
TJieils der planetarischen Sffirungen, welcher aus der Bewegung
der Sonne ensteht" (1824), in which he introduces a class of
transcendental functions, </ n (#), much used in applied mathe
matics, and known as "BessePs functions." He gave their
principal properties, and constructed tables for their eval
uation. Recently it has been observed that BessePs func
tions appear much earlier in mathematical literature. 98
Such functions of the zero order occur in papers of Daniel
Bernoulli (1732) and Euler on vibration of heavy strings sus
pended from one end. All of BessePs functions of the first
kind and of integral orders occur in paper by Euler (1764) on
the vibration of a stretched elastic membrane. In 1878 Lord
Rayleigh proved that BessePs functions are merely particular
cases of Laplace s functions. J. W. L. G-laisher illustrates
by BessePs functions his assertion that mathematical branches
growing out of, physical inquiries as a rule "lack the easy flow
APPLIED MATHEMATICS. 375
or homogeneity of form which is characteristic of a mathemati
cal theory properly so called." These functions have been
studied by C. Th. Anger of Danzig, 0. Schlomilch of Dresden,
Ku Lipschitz of Bonn (born 1832), Carl Neumann of Leipzig
(born 1832), Eugen Lommel of Leipzig, I. Todhunter of St.
John s College, Cambridge.
Prominent among the successors of Laplace are the follow
ing: Simeon Denis Poisson (1781-1840), who wrote in 1808
a classic M6moire sur les inegalites sfoulaires des moyens mouve-
ments des plan&tes. Giovanni Antonio Amadeo Plana (1781-
1864) of Turin, a nephew of Lagrange, who published in 1811
a Memoria sulla teoria dell 3 attrazione degli sferoidi ellitici, and
contributed to the theory of the moon. Peter Andreas Hansen
(1795-1874) of G-otha, at one time a clockmaker in Tondern,
then Schumacher s assistant at Altona, and finally director of
the observatory at Grotha, wrote on various astronomical sub
jects, but mainly on the lunar theory, which he elaborated in
his work Fundamenta nova investigationes orbitcB verce quam
Luna perlustrat (1838), and in subsequent investigations
embracing extensive lunar tables. George Biddel Airy (1801-
1892), royal astronomer at Greenwich, published in 1826 his
Mathematical Tracts on the Lunar and Planetary Theories.
These researches have since been greatly extended by him.
August Ferdinand Mobius (1790-1868) of Leipzig wrote, in 1842,
Elemente der Mechanik des Himmels. Urbain Jean Joseph Le
Verrier (1811-1877) of Paris wrote, the Eecherches Astrono-
miqueSj constituting in part a new elaboration of celestial
mechanics, and is famous for his theoretical discovery of
Neptune. John Couch Adams (1819-1892) of Cambridge
divided with Le Verrier the honour of the mathematical dis
covery of Neptune, and pointed out in 1853 that Laplace s
explanation of the secular acceleration of the moon s mean
motion accounted for only half the observed acceleration.
876 A HISTORY OF MATHEMATICS.
Charles Eugene Delaimay (born 1816, and drowned off Cher
bourg in 1872), professor of mechanics at the Sorbonne in
Paris, explained most of the remaining acceleration of the
moon, unaccounted for by Laplace s theory as corrected by
Adams, by tracing the effect of tidal friction, a theory
previously suggested independently by Kant, Eobert Mayer,
and William Ferrel of Kentucky. George Howard Darwin of
Cambridge (born 1845) made some very remarkable inves
tigations in 1879 on tidal friction, which trace with great
certainty the history of the moon from its origin. He has
since studied also the effects of tidal friction upon other
bodies in the solar system. Criticisms on some parts of his
researches have been made by James Nolan of Victoria. Simon
Newcomb (born 1835), superintendent of the Nautical Almanac
at Washington, and professor of mathematics at the Johns
Hopkins University, investigated the errors in Haiisen s tables
of the moon. Eor the last twelve years the main work of the
17. &. Nautical Almanac office has been to collect and discuss
data for new tables of the planets which will supplant the
tables of Le Verrier. G. W. Hill of that office has contributed
an elegant paper on certain possible abbreviations in the com
putation of the long-period of the moon s motion, due to the
direct action of the planets, and has made the most elaborate
determination yet undertaken of the inequalities of the moon s
motion due to the figure of the earth. He has also computed
certajpHkunar inequalities due to the action of Jupiter.
x Fhe mathematical discussion of Saturn s rings was taken up
first by Laplace, who demonstrated that a homogeneous solid
ring could not be in equilibrium, and in 1851 by B. Peirce,
who proved their non-solidity by showing that even an irregu
lar solid ring could not be in equilibrium about Saturn. The
mechanism of these rings was investigated by James Clerk
Maxwell in an essay to which the Adams prize was awarded.
APPLIED MATHEMATICS. 377
He concluded that they consisted of an aggregate of uncon
nected particles.
The problem of three bodies has been treated in various
ways since the time of Lagrange, but no decided advance
towards a more .complete algebraic solution has been made,
and the problem stands substantially where it was left by him.
He had made a reduction in the differential equations to the
seventh order. This was elegantly accomplished in a different
way by Jacobi in 1843. J3. Radau (Comptes Rendus, LXVIL,
1868, p. 841) and AlUgret (Journal de MatMmatiques, 1875,
p. 277) showed that the reduction can be performed on the
equations in their original form. Noteworthy transformations
and discussions of the problem have been given by J. L. IT.
Bertrand, by Emile Bour (1831-1866) of the Polytechnic School
in Paris, by Mathieu, Hesse, J. A. Serret. H. Bruns of Leipzig
has shown that no advance in the problem of three or of n
bodies may be expected by algebraic integrals, and that we
must look to the modern theory of functions for a complete
solution (Acta Math., XL, p. 43)."
Among valuable text-books on mathematical astronomy rank
the following works : Manual of Spherical and Practical Astron
omy by Chauvenet (1863), Practical and Spherical Astronomy
by Robert Main of Cambridge, TJieoretical Astronomy by James
C. Watson of Ann Arbor (1868), Traite tlementaire de M&ca-
nique Celeste of H. Eesal of the Polytechnic School in Paris,
Cours d Astronomie de VEcole PolytecJimque by Faye, Trait6
de M6canique Celeste by Tisserandj Lehrbuch der JBahnbestim-
mung by T. Oppolzer, Mathematische Theorien der Planeten-
bewegung by 0. DziobeJc, translated into English by M. W,
Harrington and W. J. Hussey.
During the present century we have come to recognise the
advantages frequently arising from a geometrical treatment of
mechanical problems. To Poinsot, Chasles, and Mobius we
378 A HISTORY OF MATHEMATICS.
owe the most important developments made in geometrical
mechanics. Louis Poinsot (1777-1859) , a graduate of the
Polytechnic School in Paris,, and for many years member of
the superior council of public instruction, published in 1804
his EUments de Statique. This work is remarkable not only
as being the earliest introduction to synthetic mechanics, but
also as containing for the first time the idea of couples, which
was applied by Poinsot in a publication of 1834 to the theory
of rotation. A clear conception of the nature of rotary
motion was conveyed by Poinsot s elegant geometrical repre
sentation by means of an ellipsoid rolling on a certain fixed
plane. This construction was extended by Sylvester so as
to measure the rate of .rotation of the ellipsoid on the plane.
A particular class of dynamical problems has recently been
treated geometrically^ by Sir Robert Stawell Ball, formerly
astronomer royal of Ireland, now Lowndean Professor of
Astronomy and Geometry at Cambridge. His method is given
in a work entitled Theory of Screws, Dublin, 1876, and in
subsequent articles. Modern geometry is here drawn upon,
as was done also by Clifford in the related subject of Bi-
quaternions. Arthur Buchheim of Manchester (1859-1888),
showed that G-rassmann s Ausdehnungslehre supplies all the
necessary materials for a simple calculus of screws in elliptic
space. Horace Lamb applied the theory of screws to the ques
tion of the steady motion of any solid in a fluid.
Advances in theoretical mechanics, bearing on the in
tegration and the alteration in form of dynamical equations,
were made since Lagrange by Poisson, William Eowan Hamil
ton., Jacobi, Madame Kowalevski, and others. Lagrange had
established the "Lagrangian form" of the equations of
motion. He had given a theory of the variation of the
arbitrary constants which, however, turned out to be less
fruitful in results than a theory advanced by Poisson." Pois-
APPLIED MATHEMATICS. 379
son s theory of the variation of the arbitrary constants and
the method of integration thereby afforded marked the first
onward step since Lagrange. Then came the researches of
Sir William Kowan Hamilton. His discovery that the inte
gration of the dynamic differential equations is connected with
the integration of a certain partial differential equation of the
first order and second degree, grew out of an attempt to deduce,
by the undulatory theory, results in geometrical optics previ
ously based on the conceptions of the emission theory. The
Philosophical Transactions of 1833 and 1834 contain Hamil
ton s papers, in which appear the first applications to me
chanics of the principle of varying action and the characteristic
function, established by him some years previously. The
object which Hamilton proposed to himself is indicated by
the title of his first paper, viz. the discovery of a function
by means of which all integral equations can be actually
represented. The new form obtained by him for the equation
of motion is a result of no less importance than that which
was the professed object of the memoir. Hamilton s method
of integration was freed by Jacobi of an unnecessary complica
tion, and was then applied by him to the determination of a
geodetic line on the general ellipsoid. With aid of elliptic co
ordinates Jacobi integrated the partial differential equation
and expressed the equation of the geodetic in form of a
relation between two Abelian integrals. Jacobi applied to
differential equations of dynamics the theory of the ultimate
multiplier. The differential equations of dynamics are only
one of the classes of differential equations considered by
Jacobi. Dynamic investigations along the lines of Lagrange,
Hamilton, and Jacobi were made by Liouville, A. Desboves,
Serret, J. C. F. Sturm, Ostrogradsky, J. Bertrand, Donkin,
Brioschi, leading up to the development of the theory of a
system of canonical integrals.
380 A HISTORY OF MATHEMATICS.
An important addition to the theory of the motion of a solid
body about a fixed point was made by Madame SopMe de
Kowalevski 96 (1853-1891), who discovered a new case in which
the differential equations of motion can be integrated. By
the use of theta-functions of two independent variables she
furnished a remarkable example of how the modern theory of
functions may become useful in mechanical problems. She
was a native of Moscow, studied under Weierstrass, obtained
the doctor s degree at Gottingen, and from 1884 until her
death was professor of higher mathematics at the University
of Stockholm. The research above mentioned received the
Bordin prize of the French Academy in 1888, which was
doubled on account of the exceptional merit of the paper.
There are in vogue three forms for the expression of the
kinetic energy of a dynamical system: the Lagrangian, the
Hamiltonian, and a modified form of Lagrange s equations in
which, certain velocities are omitted. The kinetic energy
is expressed in the first form as a homogeneous quadratic
function of the velocities, which are the time-variations of the
co-ordinates of the system; in the second form, as a homo
geneous quadratic function of the momenta of the system;
the third form, elaborated recently by Edward John Eouth
of Cambridge, in connection with his theory of " ignoration of
co-ordinates," and by A. B. Basset, is of importance in hydro-
dynamical problems relating to the motion of perforated solids
in a liquid, and in other branches of physics.
In recent time great practical importance has come to be
attached to the principle of mechanical similitude. By it one
can determine from the performance of a model the action of
the machine constructed on a larger scale. The principle was
first enunciated by Newton (Principia, Bk. II., Sec. VIII.,
Prop. 32), and was derived by Bertrand from the principle
of virtual velocities. A corollary to it, applied in ship-build- %
APPLIED MATHEMATICS. 381
ing, goes by the name of William Ifroude s law, but was enun
ciated also by Heech.
The present problems of dynamics differ materially from
those of the last century. The explanation of the orbital and
axial motions of the heavenly bodies by the law of universal
gravitation was the great problem solved by Clairaut, Euler,
D Alembert, Lagrange, and Laplace. It did not involve the
consideration of frictional resistances. In the present time
the aid of dynamics has been invoked by the physical
sciences. The problems there arising are often complicated
by the presence of friction. Unlike astronomical problems of
a century ago, they refer to phenomena of matter and motion
that are usually concealed from direct observation. The great
pioneer in such problems is Lord Kelvin. While yet an
undergraduate at Cainb ridge, during holidays spent at the
seaside, he entered upon researches of this kind by working
out the theory of spinning tops, which previously had been
only partially explained by Jellet in his Treatise on the Tlieory
of Friction (1872), and by Archibald Smith.
Among standard works on mechanics are Jacobi s Vorlesun-
gen uber Dynamite, edited by Clebseh, 1866 ; KirchliolFs Vorle-
sungen uber mathematische PhysiJc, 1876 ; Benjamin Peirce s
Analytic Mechanics, 1855; SomofPs TheoretiscJie MechaniJc,
1879; Tait and Steele s Dynamics of a Particle, 1856; Minchin s
Treatise on Statics; Routh s Dynamics of a System of Rigid
Bodies; Sturm s Cours de M&canique de VEcole Polytechnique.
The equations which constitute the foundation of the theory
of fluid motion were fully laid down at the time of Lagrange,
but the solutions actually worked out were few and mainly
of the irrotational type. A powerful method of attacking
problems in fluid motion is that of images, introduced in 1843
by George Gabriel Stokes of Pembroke College, Cambridge.
Tt received little attention until Sir William Thomson s dis-
382 A HISTORY OF MATHEMATICS.
covery of electrical images, whereupon the theory was extended
by Stokes, Hicks, and Lewis. In 1849, Thomson gave the
maximum and minimum theorem peculiar to hydrodynamics,
which was afterwards extended to dynamical problems in
general.
A new epoch in the progress of hydrodynamics was created,
in 1856, by Helmholtz, who worked out remarkable properties
of rotational motion in a homogeneous, incompressible fluid,
devoid of viscosity. He showed that the vortex filaments in
such a medium may possess any number of knottings and twist-
ings, but are either endless or the ends are in the free surface
of the medium ; they are indivisible. These results suggested
to Sir William Thomson the possibility of founding on them a
new form of the atomic theory, according to which every atom
is a vortex ring in a non-frictional ether, and as such must be
absolutely permanent in substance and duration. The vortex-
atom theory is discussed by J. J. Thomson of Cambridge
(born 1856) in his classical treatise on the Motion of Vortex
Rings, to which the Adams Prize was awarded in 1882.
Papers on vortex motion have been published also by Horace
Lamb, Thomas Craig, Henry A. Eowland, and Charles Chree.
The subject of jets was investigated by Helmholtz, ELirch-
hoff, Plateau, and Rayleigh ; the motion of fluids in a fluid by
Stokes, Sir W. Thomson, Kopcke, G-reenhill, and Lamb ; the
theory of viscous fluids by Navier, Poisson, Saint- Yenant,
Stokes, 0. E. Meyer, Stefano, Maxwell, Lipschitz, Craig,
Helmholtz, and A. B. Basset. Viscous fluids present great
difficulties, because the equations of motion have riot the same
degree of certainty as in perfect fluids, on account of a defi
cient theory of friction, and of the difficulty of connecting
oblique pressures on a small area with the differentials of the
velocities.
Waves in liquids have been a favourite subject with Eng-
APPLIED MATHEMATICS. 383
lisli mathematicians. The early inquiries of Poisson and
Cauchy were directed to the investigation of waves produced
by disturbing causes acting arbitrarily on a small portion
of the fluid. The velocity of the long wave was given
approximately by Lagrange in 1786 in case of a channel of
rectangular cross-section, by Green in 1839 for a channel of
triangular section, and by P. Kelland for a channel of any
uniform section. Sir George B. Airy, in his treatise on Tides
and Waves, discarded mere approximations, and gave the exact
equation on which the theory of the long wave in a channel of
uniform rectangular section depends. But he gave no general
solutions. J". McCowan of University College at Dundee
discusses this topic more fully, and arrives at exact and
complete solutions for certain cases. The most important
application of the theory of the long wave is to the explana
tion of tidal phenomena in rivers and estuaries.
The mathematical treatment of solitary waves was first
taken up by S. Earnshaw in 1845, then by Stokes ; but the first
sound approximate theory was given by J. Boussinesq in 1871,
who obtained an equation for their form, and a value for the
velocity in agreement with experiment. Other methods of
approximation were given by Eayleigh and J. McCowan. In
connection with deep-water waves, Osborne Reynolds gave in
1877 the dynamical explanation for the fact that a group
of such waves advances with only half the rapidity of the
individual waves.
The solution of the problem of the general motion of an
ellipsoid in a fluid is due to the successive labours of Green
(1833), Clebsch (1856), and Bjerknes (1873). The free
motion of a solid in a liquid has been investigated by W.
Thomson, Kirchhoff, and Horace Lamb. By these labours, the
motion of a single solid in a fluid has come to be pretty well
understood, but the case of two solids in a fluid is not devel-
384 A HISTORY OF MATHEMATICS.
oped so fully. The problem has been attacked by W. M.
Hicks.
The determination of the period of oscillation of a rotating
liquid spheroid has important bearings on the question of the
origin of the moon. G-. H. Darwin s investigations thereon,
viewed in the light of Eiemann s and Poincare s researches,
seem to disprove Laplace s hypothesis that the moon separated
from the earth as a ring, because the angular velocity was too
great for stability ; Darwin finds no instability.
The explanation of the contracted vein has been a point of
much controversy, but has been put in a much better light by
the application of the principle of momentum, originated by
Eroude and Eayleigh. Eayleigh considered also the reflection
of waves, not at the surface of separation of two uniform
media, where the transition is abrupt, but at the confines of
two media between which the transition is gradual.
The first serious study of the circulation of winds on the
earth s surface was instituted at the beginning of the second
quarter of this century by H. W. Dov } William (7. JZedJield, and
James P. Espy, followed by researches of W. Reid, Piddington,
and JSlias Loomis. But the deepest insight into the wonder
ful correlations that exist among the varied motions of the
atmosphere was obtained by William Ferrel (1817-1891). He
was born in Fulton County, Pa., and brought up on a farm.
Though in unfavourable surroundings, a burning thirst for
knowledge spurred the boy to the mastery of one branch after
another. He attended Marshall College, Pa., and graduated
in 1844 from Bethany College. While teaching school he
became interested in meteorology and in the subject of tides.
In 1856 he wrote an article on " the winds and currents of the
ocean." The following year he became connected with the
Nautical Almanac. A mathematical paper followed in 1858
on "the motion of fluids and solids relative to the earth s
APPLIED MATHEMATICS. 385
surface. " The subject was extended afterwards so as to
embrace the mathematical theory of cyclones, tornadoes,
water-spouts, etc. In 1885 appeared his Recent Advances in
Meteorology. In the opinion of a leading European meteor
ologist (Julius Hann of Vienna), Ferrel has "contributed more
to the advance of the physics of the atmosphere than any
other living physicist or meteorologist."
Ferrel teaches that the air flows in great spirals toward the
poles, both in the upper strata of the atmosphere and on the
earth s surface beyond the 30th degree of latitude; while
the return current blows at nearly right angles to the above
spirals, in the middle strata as well as on the earth s surface,
in a zone comprised between the parallels 30 IsT. and 30 S. The
idea of three superposed currents blowing spirals was first
advanced by James Thomson, but was published in very
meagre abstract.
FerrePs views have given a strong impulse to theoretical
research in America, Austria, and Germany. Several objec
tions raised against his argument have been abandoned, or
have been answered by "W. M. Davis of Harvard. The mathe
matical analysis of F. Waldo of Washington, and of others,
has further confirmed the accuracy of the* theory. The trans
port of Krakatoa dust and observations made on clouds point
toward the existence of an upper east current on the equator,
and Pernter has mathematically deduced from FerrePs theory
the existence of such a current.
Another theory of the general circulation of the atmosphere
was propounded by Werner Siemens of Berlin, in which an
attempt is made to apply thermodynamics to aerial currents.
Important new points of view have been introduced recently
by Helmholtz, who concludes that when two air currents blow
one above the other in different directions, a system of air
waves must arise in the same way as waves are formed on the
386 A HISTOKY OF MATHEMATICS.
sea. He and A. Oberbeck showed that when the waves on the
sea attain lengths of from 16 to 33 feet, the air waves must
attain lengths of from 10 to 20 miles, and proportional depths.
Superposed strata would thus mix more thoroughly, and their
energy would be partly dissipated. From hydrodynainical
equations of rotation Helrnholtz established the reason why
the observed velocity from equatorial regions is much less in
a latitude of, say, 20 or 30, than it would be were the move
ments unchecked.
About 1860 acoustics began to be studied with renewed
zeal. The mathematical theory of pipes and vibrating strings
had been elaborated in the eighteenth century by Daniel Ber
noulli, D Alembert, Euler, and Lagrange. In the first part of
the present century Laplace corrected Newton s theory on the
velocity of sound in gases, Poisson gave a mathematical dis
cussion of torsional vibrations ; Poisson, Sophie Germain, and
Wheatstone studied Chladni s figures ; Thomas Young and the
brothers Weber developed the wave-theory of sound. Sir J.
F. W. Herschel wrote on the mathematical theory of sound for
the Encydopc&dia, Metropolitana, 1845. Epoch-making were
Helmholtz s experimental and mathematical researches. In
his hands and Rayleigh s, Fourier s series received due
attention. Helmholtz gave the mathematical theory of beats,
difference tones, and summation tones. Lord Rayleigh (John
William Strutt) of Cambridge (born 1842) made extensive
mathematical researches in acoustics as a part of the theory of
vibration in general. Particular mention may be made of his
discussion of the disturbance produced by a spherical obstacle
on the waves of sound, and of phenomena, such as sensitive
flames, connected with the instability of jets of fluid. In 1877
and 1878 he published in two volumes a treatise on TJie Theory
of Sound. Other mathematical researches on this subject have
been made in England by Donkin and Stokes.
APPLIED MATHEMATICS. 387
The theory of elasticity 42 belongs to this century. Before
1800 no attempt had been made to form general equations for
the motion or equilibrium of an elastic solid. Particular prob
lems had been solved by special hypotheses. Thus, James
Bernoulli considered elastic laminae; Daniel Bernoulli and
Euler investigated vibrating rods; Lagrange and Euler, the
equilibrium of springs and columns. The earliest investiga
tions of this century, by Thomas Young (" Young s modulus of
elasticity ") in England, J. Binet in France, and Gr. A. A. Plana
in Italy, were chiefly occupied in extending and correcting the
earlier labours. Between 1830 and 1840 the broad outline of the
modern theory of elasticity was established. This was accom
plished almost exclusively by French writers, Louis-Marie-
Henri JSTavier (1785-1836), Poisson, Cauchy, Mademoiselle
Sophie Germain (1776-1831), Felix Savart (1791-1841).
Simeon Denis Poisson 94 (1781-1840) was born at Pithiviers.
The boy was put out to a nurse, and he used to tell that when
his father (a common soldier) came to see him one day, the
nurse had gone out and left him suspended by a thin cord to a
nail in the wall in order to protect him from perishing under
the teeth of the carnivorous and unclean animals that roamed
on the floor. Poisson used to add that his gymnastic efforts
when thus siispended caused him to swing back and forth, and
thus to gain an early familiarity with the pendulum, the study
of which occupied him much in his maturer life. His father
destined him for the medical profession, but so repugnant was
this to him that he was permitted to enter the Polytechnic
School at the age of seventeen. His talents excited the inter
est of Lagrange and Laplace. At eighteen he wrote a memoir
on finite differences which was printed on the recommendation
of Legendre. He soon became a lecturer at the school, and
continued through life to hold various government scientific
posts and professorships. He prepared some 400 publications,
388 A HISTORY OF MATHEMATICS.
mainly on applied mathematics. His Traite de Mfaanique,
2 vols., 1811 and 1833, was long a standard work. He wrote
on the mathematical theory of heat, capillary action, proba
bility of judgment, the mathematical theory of electricity and
magnetism, physical astronomy, the attraction of ellipsoids,
definite integrals, series, and the theory of elasticity. He was
considered one of the leading analysts of his time.
His work on elasticity is hardly excelled by that of Cauchy,
and second only to that of Saint-Venant. There is hardly a
problem in elasticity to which he has not contributed, while
many of his inquiries were new. The equilibrium and motion
of a circular plate was first successfully treated by him.
Instead of the definite integrals of earlier writers, he used
preferably finite summations. Poisson s contour conditions
for elastic plates were objected to by Gustav Kirehhoff of
Berlin, who established new conditions. But Thomson and
Tait in their Treatise on Natural Philosophy have explained
the discrepancy between Poisson s and KirchhofPs boundary
conditions, and established a reconciliation between them. .
Important contributions to the theory of elasticity were
made by Cauchy. To him we owe the origin of the theory
of stress, and the transition from the consideration of the
force upon a molecule exerted by its neighbours to the con
sideration of the stress upon a small plane at a point. He
anticipated Green and Stokes in giving the equations of iso-
tropic elasticity with two constants. The theory of elasticity
was presented by Gabrio Piola of Italy according to the prin
ciples of Lagrange s Mtcanique Analytique, but the superiority
of this method over that of Poisson and Cauchy is far from
evident. The influence of temperature on stress was first
investigated experimentally by Wilhelm Weber of Gottingen,
and afterwards mathematically by Duhamel, who, assuming
Poisson s theory of elasticity, examined the alterations of
APPLIED MATHEMATICS. 389
form which the formulae undergo when we allow for changes
of temperature. Weber was also the first to experiment
on elastic after-strain. Other important experiments were
made by different scientists, which disclosed a wider range
of phenomena^ and demanded a more comprehensive theory.
Set was investigated by Gerstner (1756-1832) and Eaton
Hodgkinson, while the latter physicist in England and Yicat
(1786-1861) in Prance experimented extensively on absolute
strength. Vicat boldly attacked the mathematical theories of
flexure because they failed to consider shear and the time-ele
ment. As a result, a truer theory of flexure was soon pro
pounded by Saint-Venant. Poncelet advanced the theories of
resilience and cohesion.
Gabriel Lame 94 (1795-1870) was born at Tours, and gradu
ated at the Polytechnic School. He was called to Russia
with Clapeyron and others to superintend the construction of
bridges and roads. On his return, in 1832, he was elected
professor of physics at the Polytechnic School. Subsequently
he held various engineering posts and professorships in Paris.
As engineer he took an active part in the construction of the
first railroads in Prance. Lame devoted his fine mathemati
cal talents mainly to mathematical physics. In four works :
Legons sur les fonctions inverses des transcendantes et Us sur
faces isothermes; Sur les coordonnees curvilignes et leurs diver ses
applications; Sur la theorle analytique de la clialeur; Sur la
tMorie math&matique de V elasticity des corps solides (1852) , and
in various memoirs he displays fine analytical powers ; but a
certain want of physical touch sometimes reduces the value of
his contributions to elasticity and other physical subjects. In
considering the temperature in the interior of an ellipsoid
under certain conditions, he employed functions analogous to
Laplace s functions, and known by the name of " Lame s func
tions." A problem in elasticity called by Lame s name, viz.
390 A HISTOEY OF MATHEMATICS.
to investigate the conditions for equilibrium of a spherical
elastic envelope subject to a given distribution of load on
the bounding spherical surfaces, and the determination of the
resulting shifts is the only completely general problem on
elasticity which can be said to be completely solved. He
deserves much credit for his derivation and transformation
of the general elastic equations, and for his application of
them to double refraction. Rectangular and triangular mem
branes were shown by him to be connected with questions in
the theory of numbers. The field of photo-elasticity was
entered upon by Lame, E. E. Neumann, Clerk Maxwell.
Stokes, Wertheim, E. Clausius, Jellett, threw new light upon
the subject of "raii-constancy" and "multi-constancy," which
has long divided elasticians into two opposing factions. The
uni-constant isotropy of Navier and Poisson had been ques
tioned by Cauchy, and was now severely criticised by Green
and Stokes.
Barre de Saint-Venant (1797-1886), ingenieur des ponts et
chaussees, made it his life-work to render the theory of
elasticity of practical value. The charge brought by practical
engineers, like Vicat, against the theorists led Saint-Venant to
place the theory in its true place as a guide to the practical
man. Numerous errors committed by his predecessors were
removed. He corrected the theory of flexure by the considera
tion of slide, the theory of elastic rods of double curvature by
the introduction of the third moment, and the theory of tor
sion by the discovery of the distortion of the primitively
plane section. His results on torsion abound in beautiful
graphic illustrations. In case of a rod, upon the side surfaces
of which no forces act, he showed that the problems of flexure
and torsion can be solved, if the end-forces are distributed
over the end-surfaces by a definite law. Clebsch, in his
Lehrbuch der Elasticitat, 1862, showed that this problem is
APPLIED MATHEMATICS. 391
reversible to the case of side-forces without end-forces.
Clebsch 68 extended the research to very thin rods and to very
thin plates. Saint-Yenant considered problems arising in the
scientific design of built-up artillery, and his solution of them
differs considerably from Lame s solution, which was popular
ised by Rankine, and much used by gun-designers. In Saint-
Venant s translation into French of Clebsch s Elasticitat, he
develops 4 extensively a double-suffix notation for strain and
stresses. Though often advantageous., this notation is cum
brous, and has not been generally adopted. Karl Pearson,
professor in University College, London, has recently exam
ined mathematically the permissible limits of the application
of the ordinary theory of flexure of a beam.
The mathematical theory of elasticity is still in an unsettled
condition. Not only are scientists still divided into two
schools of " rari-constancy " and " multi-constancy," but differ
ence of opinion exists on other vital questions. Among the
numerous modern writers on elasticity may be mentioned
Entile Mathieu (1835-1891), professor at Besaneon, Maurice
Levy of Paris, Charles Chree, superintendent of the Kew Ob
servatory, A. B. Basset, Sir William Thomson (Lord Kelvin)
of Glasgow, J. Boussinesq of Paris, and others. Sir William
Thomson applied the laws of elasticity of solids to the investi
gation of the earth s elasticity, which is an important element
in the theory of ocean-tides. If the earth is a solid, then its
elasticity co-operates with gravity in opposing deformation
due to the attraction of the sun and moon. Laplace had
shown how the earth would behave if it resisted deformation
only by gravity. Lam6 had investigated how a solid sphere
would change if its elasticity only came into play. Sir
William Thomson combined the two results, and compared
them with the actual deformation. Thomson, and afterwards
G-. H. Darwin, computed that the resistance of the earth to
392 A HISTOKY OF MATHEMATICS.
tidal deformation is nearly as great as though it were of steel.
This conclusion has been confirmed recently by Simon NQW-
comb, from the study of the observed periodic changes in
latitude. For an ideally rigid earth the period would be 360
days, but if as rigid as steel, it would be 441, the observed
period being 430 days.
Among text-books on elasticity may be mentioned the works
of Lame, Clebsch, Winkler, Beer, Mathieu, W. J. Ibbetson, and
F. Neumann, edited by 0. B. Meyer.
Riemann s opinion that a science of physics only exists since
the invention of differential equations finds corroboration even
in this brief and fragmentary outline of the progress of mathe
matical physics. The undulatory theory of light, first ad
vanced by Huygens, owes much to the power of mathematics :
by mathematical analysis its assumptions were worked out
to their last consequences. Thomas Young 95 (1773-1829) was
the first to explain the principle of interference, both of
light and sound, and the first to bring forward the idea
of transverse vibrations in light waves. Young s explana
tions, not being verified by him by extensive numerical calcu
lations, attracted little notice, and it was not until Augustin
Fresnel (1788-1827) applied mathematical analysis to a much
greater extent than Young had done, that the undulatory
theory began to carry conviction. Some of Fresnel s mathe
matical assumptions were not satisfactory; hence Laplace,
Poisson, and others belonging to the strictly mathematical
school, at first disdained to consider the theory. By their
opposition Fresnel was spurred to greater exertion. Arago
was the first great convert made by Fresnel. When polarisa
tion and double refraction were explained by Young and
Fresnel, then Laplace was at last won over. Poisson drew
from Fresnel s formulae the seemingly paradoxical deduction
that a small circular disc, illuminated by a luminous point,
APPLIED MATHEMATICS. 398
must cast a shadow with, a bright spot in the centre. But
this "was found to be in accordance with fact. The theory
was taken up by another great mathematician, Hamilton, who
from his formulae predicted conical refraction, verified experi
mentally by Lloyd. These predictions do not prove, however,
that Fresnel s formulae are correct, for these prophecies might
have been made by other forms of the wave-theory. The
theory was placed on a sounder dynamical basis by the writ
ings of Cauchy, Biot, Green, C. Neumann, Elrchhoff, McCullagh,
Stokes, Saint-Venant, Sarrau, Lorenz, and Sir William Thom
son. In the wave-theory, as taught by Green and others, the
luminiferous ether was an incompressible elastic solid, for
the reason that fluids could not propagate transverse vibra
tions. But, according to Green, such an elastic solid would
transmit a longitudinal disturbance with infinite velocity.
Stokes remarked, however, that the ether might act like
a fluid in case of finite disturbances, and like an elastic solid
in case of the infinitesimal disturbances in light propagation.
Presnel postulated the density of ether to be different in
different media, but the elasticity the same, while C. Neumann
and McCullagh assume the density uniform and the elasticity
different in all substances. On the latter assumption the
direction of vibration lies in the plane of polarisation, and not
perpendicular to it, as in the theory of Eresnel.
While the above writers endeavoured to explain all optical
properties of a medium on the supposition that they arise
entirely from difference in rigidity or density of the ether in
the medium, there is another school advancing theories in
which the mutual action between the molecules of the body
and the ether is considered the main cause of refraction and
dispersion. 100 The chief workers in this field are J. Boussinesq,
W. Sellmeyer, Helrnholtz, E. Lommel, E. Ketteler, W. Voigt,
and Sir William Thomson in his lectures delivered at the
394 A. HISTORY OE MATHEMATICS.
Johns Hopkins University in 1884. Neither this nor the
first-named school succeeded in explaining all the phenomena.
A third school was founded by Maxwell. He proposed the
electro-magnetic theory, which has received extensive develop
ment recently. It will be mentioned again later. According
to Maxwell s theory, the direction of vibration does not lie
exclusively in the plane of polarisation, nor in a plane perpen
dicular to it, but something occurs in both planes a magnetic
vibration in one, and an electric in the other. Fitzgerald and
Trouton in Dublin verified this conclusion of Maxwell by
experiments on electro-magnetic waves.
Of recent mathematical and experimental contributions to
optics, mention must be made of H. A. Rowland s theory of
concave gratings, and of A. A. Michelson s work on interfer
ence, and his application of interference methods to astro
nomical measurements.
In electricity the mathematical theory and the measure
ments of Henry Cavendish (1731-1810), and in magnetism
the measurements of Charles Augustin Coulomb (1736-1806),
became the foundations for a system of measurement. For
electro-magnetism the same thing was done by Andre Marie
Ampere (1775-1836). The first complete method of measure
ment was the system of absolute measurements of terrestrial
magnetism introduced by Gauss and Wilhelm Weber (1804-
1891) and afterwards extended by Wilhelm Weber and F.
Kohlrausch to electro-magnetism and electro-statics. In 1861
the British Association and the Royal Society appointed a
special commission with Sir William Thomson at the head, to
consider the unit of electrical resistance. The commission
recommended a unit in principle like W. Weber s, but greater
than Weber s by a factor of 10 7 . 101 The discussions and labours
an this subject continued for twenty years, until in 1881 a
general agreement was reached at an electrical congress in Paris.
APPLIED MATHEMATICS. 395
A function of fundamental importance in the mathematical
theories of electricity and magnetism is the " potential." It
was first used by Lagrange in the determination of gravita
tional attractions in 1773. Soon after, Laplace gave the
celebrated differential equation,
, , =0
da? dy 2 dz*
which was extended by Poisson by writing 4?rfc in place of
zero in the right-hand member of the equation, so that it
applies not only to a point external to the attracting mass,
but to any point whatever. The first to apply the potential
function to other than gravitation problems was George Green
(1793-1841). He introduced it into the mathematical theory
of electricity and magnetism. Green was a self-educated man
who started out as a baker, and at his death was fellow of
Caius College, Cambridge. In 1828 he published by subscrip
tion at Nottingham a paper entitled Essay on the application
of mathematical analysis to the theory of electricity and magne
tism. It escaped the notice even of English mathematicians
until 1846, when Sir William Thomson had it reprinted in
Crelle s Journal, vols. xliv. and xlv. It contained what is now
known as " Green s theorem " for the treatment of potential.
Meanwhile all of Green s general theorems had been re-dis-
coverecl by Sir William Thomson, Chasles, Sturm, and Gauss.
The term potential function is due to Green. Hamilton used
the word force-function, while Gauss, who about 1840 secured
the general adoption of the function, called it simply potential
Large contributions to electricity and magnetism have been
made by William Thomson. He was born in 1824 at Belfast,
Ireland, but is of Scotch descent. He and his brother James
studied in Glasgow. Prom there he entered Cambridge, and
was graduated as Second Wrangler in 1845. William Thorn-
896 A HISTOBY OF MATHEMATICS.
son, Sylvester, Maxwell, Clifford, and J. J. Thomson are a group
of great men who were Second Wranglers at Cambridge. At
the age of twenty-two W. Thomson was elected professor of
natural philosophy in the University of Glasgow, a position
which he has held ever* since. For his brilliant mathematical
and physical achievements he was knighted, and in 1892 was
made Lord Kelvin. His researches on the theory of potential
are epoch-making. What is called "Dirichlet s principle"
was discovered by him in 1848, somewhat earlier than by
Dirichlet. We owe to Sir William Thomson new synthetical
methods of great elegance, viz. the theory of electric images
and the method of electric inversion founded thereon. By
them he determined the distribution of electricity on a bowl,
a problem previously considered insolvable. The distribution
of static electricity on conductors had been studied before this
mainly by Poisson and Plana. In 1845 F. E. Neumann of
Konigsberg developed from the experimental laws of Lenz the
mathematical theory of magneto-electric induction. In 1855
W. Thomson predicted by mathematical analysis that the dis
charge of a Leyden jar through a linear conductor would in
certain cases consist of a series of decaying oscillations. This
was first established experimentally by Joseph Henry of
Washington. William Thomson worked out the electro-static
induction in submarine cables. The subject of the screening
effect against induction, due to sheets of different metals, was
worked out mathematically by Horace Lamb and also by
Charles Niven. W. Weber s chief researches were on electro
dynamics. Helmholtz in 1851 gave the mathematical theory
of the course of induced currents in various cases. Gustav
Robert Zirchlioff w (1824-1887) investigated the distribution of
a current over a flat conductor, and also the strength of current
in each branch of a network of linear conductors.
The entire subject of electro-magnetism was revolutionised
APPLIED MATHEMATICS. 397
by James Clerk Maxwell (1831-1879). He was bom near
Edinburgh, entered the University of Edinburgh, and became
a pupil of Kelland and Forbes. In 1850 he went to Trinity
College, Cambridge, and came out Second Wrangler, E. Eouth
being Senior Wrangler. Maxwell then became lecturer at
Cambridge, in 1856 professor at Aberdeen, and in 1860
professor at King s College, London. In 1865 he retired to
private life until 1871, when he became professor of physics
at Cambridge. Maxwell not only translated into mathematical
language the experimental results of Faraday, but established
the electro-magnetic theory of light, since verified experimen
tally by Hertz. His first researches thereon were published
in 1864. In 1871 appeared his great Treatise on Electricity
and Magnetism. He constructed the electro-magnetic theory
from general equations, which are established upon purely
dynamical principles, and which determine the state of the
electric field. It is a mathematical discussion of the stresses
and strains in a dielectric medium subjected to electro-magnetic
forces. The electro-magnetic theory has received developments
from Lord Eayleigh, J. J. Thomson, H. A. Eowland, E. T.
Glazebrook, H. Helmholtz, L. Boltzmann, 0. Heaviside, J. H.
Poynting, and others. Hermann von Helmlioltz turned his
attention to this part of the subject in 1871. He was born
in 1821 at Potsdam, studied at the University of Berlin, and
published in 1847 his pamphlet Ueber die Erhaltung der Kraft.
He became teacher of anatomy in the "Academy of Art in
Berlin. He was elected professor of physiology at Konigs-
berg in 1849, at Bonn in 1855, at Heidelberg in 1858, It was
at Heidelberg that he produced his work on Tonempfindwng.
In 1871 he accepted the chair of physics at the University of
Berlin. From this time on he has been engaged chiefly on
inquiries in electricity and hydrodynamics. Helmholtz aimed
to determine in what direction experiments should be made to
398 A HISTORY OF MATHEMATICS.
decide between the theories of W. Weber, E. E. Neumann,
Riemann, and Clausius, who had attempted to explain electro-
dynamic phenomena by the assumption of forces acting at a dis
tance between two portions of the hypothetical electrical fluid,
the intensity being dependent not only on the distance, but also
on the velocity and acceleration, and the theory of Faraday
and Maxwell, which discarded action at a distance and assumed
stresses and strains in the dielectric. His experiments favoured
the British theory. He wrote on abnormal dispersion, and
created analogies between electro-dynamics and hydrody
namics. Lord Eayleigh compared electro-magnetic problems
with their mechanical analogues, gave a dynamical theory of
diffraction, and applied Laplace s coefficients to the theory of
radiation. Eowland made some emendations on Stokes paper
on diffraction and considered the propagation of an arbitrary
electro-magnetic disturbance and spherical waves of light.
Electro-magnetic induction has been investigated mathemati
cally by Oliver Heaviside, and he showed that in a cable it is
an actual benefit. Heaviside and Poynting have reached
remarkable mathematical results in their interpretation and
development of Maxwell s theory. Most of Heaviside s papers
have been published since 1882 ; they cover a wide field.
One part of the theory of capillary attraction, left defective
by Laplace, namely, the action of a solid upon a liquid, and
the mutual action between two liquids, was made dynamically
perfect by Gauss. He stated the rule for "angles of contact
between liquids and solids. A similar rule for liquids was
established by Ernst Eranz Neumann. Chief among recent
workers on the mathematical theory of capillarity are Lord
Hayleigh and E. Mathieu.
The great principle of the conservation of energy was
established by Robert Mayer (1814-1878), a physician in
Heilbronn, and again independently by Colding of Copen-
APPLIED MATHEMATICS. 399
hagen, Joule, and Helmholtz. James Prescott Joule (1818-
1889) determined experimentally the mechanical equivalent
of heat. Helmholtz in 1847 applied the conceptions of the
transformation and conservation of energy to the various
branches of physics, and thereby linked together many well-
known phenomena. These labours led to the abandonment
of the corpuscular theory of heat. The mathematical treat
ment of thermic problems was demanded by practical con
siderations. Thermodynamics grew out of the attempt to
determine mathematically how much work can be gotten out
of a steam engine. Sadi-Carnot, an adherent of the corpuscular
theory, gave the first impulse to this. The principle known
by his name was published in 1824. Though the importance
of his work was emphasised by B. P. E. Clapeyron, it did not
meet with general recognition until it was brought forward
by William Thomson. The latter pointed out the necessity
of modifying Carnot s reasoning so as to bring it into accord
with the new theory of heat. William Thomson showed in
1848 that Carnot s principle led to the conception of an
absolute scale of temperature. In 1849 he published "an
account of Carnot s theory of the motive power of heat, with
numerical results deduced from B-egnault s experiments." In
February, 1850, Rudolph Clausius (1822-1888), then in Zurich,
(afterwards professor in Bonn), communicated to the Berlin
Academy a paper on the same subject which contains the
Protean second law of thermodynamics. In the same month
William John M. Rankine (1820-1872), professor of engineer
ing and mechanics at Glasgow, read before the Eoyal Society
of Edinburgh a paper in which he declares the nature of
heat to consist in the rotational motion of molecules, and
arrives at some of the results reached previously by Clausius.
He does not mention the second law of thermodynamics, but
in a subsequent paper he declares that it could be derived
400 A BISTORT OF MATHEMATICS.
from equations contained in Ms first paper. His proof of
the second law is not free from objections. In March, 1851,
appeared a paper of William Thomson which contained a
perfectly rigorous proof of the second law. He obtained it
before he had seen the researches of Clausius.- The state
ment of this law, as given by Clausius,, has been much
criticised, particularly by Eankine, Theodor Wand, P. G-.
Tait, and Tolver Preston. Eepeated efforts to deduce it from
general mechanical principles have remained fruitless. The
science of thermodynamics was developed with great suc
cess by Thomson, Clausius, and Eankine. As early as 1852
Thomson discovered the law of the dissipation of energy,
deduced at a later period also by Clausius. The latter desig
nated the non-transformable energy by the name entropy,
and then stated that the entropy of the universe tends
toward a maximum.. ITor entropy Eankine used the term
thermodynamic function. Thermodynamic investigations have
been carried on also by G. Ad. Him of Colmar, and Helm-
holtz (monocyclic and polycyclic systems). Valuable graphic
methods for the study of thermodynamic relations were de
vised in 1873-1878 by J. Willard Gibbs of Yale College.
Gibbs first gives an account of the advantages of using
various pairs of the five fundamental thermodynamic quanti
ties for graphical representation, then discusses the entropy-
temperature and entropy-volume diagrams, and the volume-
energy-entropy surface (described in Maxwell s Theory of
Heat). Gibbs formulated the energy-entropy criterion of
equilibrium and stability, and expressed it in a form appli
cable to complicated problems of dissociation. Important
works on thermodynamics have been prepared by Clausius
in 1875, by E. Euhlmann in 1875, and by Poincare in 1892.
In the study of the law of dissipation of energy and the
principle of least action, mathematics and metaphysics met on"
APPLIED MATHEMATICS. 401
common ground. The doctrine of least action was first pro
pounded by Maupertius in 1744. Two years later he pro
claimed it to be a universal law of nature, and the first
scientific proof of the existence of God. It was weakly sup
ported by him, violently attacked by Konig of Leipzig, and
keenly defended by Euler. Lagrange s conception of the prin
ciple of least action became the mother of analytic mechanics,
but his statement of it was inaccurate, as has been remarked
by Josef Bertrand in the third edition of the Mcanique Ana-
lytique. The form of the principle of least action, as it now
exists, was given by Hamilton, and was extended to electro
dynamics by F. E. Neumann, Clausius, Maxwell, and Helrn-
holtz. To subordinate the principle to all reversible processes,
Helmholtz introduced into it the conception of the "kinetic
potential. " In this form the principle has universal validity.
An offshoot of the mechanical theory of heat is the modern
kinetic theory of gases, developed mathematically by Clausius,
Maxwell) Ludivig Boltzmann of Munich, and others. The first
suggestions of a kinetic theory of matter go back as far as the
time of the Greeks. The earliest work to be mentioned here is
that of Daniel Bernoulli, 1738. He attributed to gas-molecules
great velocity, explained the pressure of a gas by molecular
bombardment, and deduced Boyle s law as a consequence of
his assumptions. Over a century later his ideas were taken
up by Joule (in 1846), A. K. Kronig (in 1856), and Clausius
(in 1857). Joule dropped his speculations on this subject
when he began his experimental work on heat. Kronig
explained by the kinetic theory the fact determined experi
mentally by Joule that the internal energy of a gas is not
altered by expansion when no external work is done. Clausius
took an important step in supposing that molecules may have
rotary motion, and that atoms in a molecule may move rela
tively to each other. He assumed that the force acting
402 A HISTOBY OF MATHEMATICS.
between molecules is a function of their distances, that tem
perature depends solely upon the kinetic energy of molecular
motions, and that the number of molecules which at any
moment are so near to each other that they perceptibly influ
ence each other is comparatively so small that it may be
neglected. He calculated the average velocities of molecules,
and explained evaporation. Objections to his theory, raised
by Buy s-Ballot and by Jochniann, were satisfactorily answered
by Clausius and Maxwell, except in one case where an addi
tional hypothesis had to be made. Maxwell proposed to him
self the problem to determine the average number of molecules,
the velocities of which lie between given limits. His expres
sion therefor constitutes the important law of distribution of
velocities named after him. By this law the distribution of
molecules according to their velocities is determined by the
same formula (given in the theory of probability) as the dis
tribution of empirical observations according to the magnitude
of their errors. The average molecular velocity as deduced
by Maxwell differs from that of Clausius by a constant factor.
Maxwell s first deduction of this average from his law of dis
tribution was not rigorous. A sound derivation was given by
0. E. Meyer in 1866. Maxwell predicted that so long as
Boyle s law is true, the coefficient of viscosity and the coeffi
cient of thermal conductivity remain independent of the press
ure. His deduction that the coefficient of viscosity should
be proportional to the square root of the absolute temperature
appeared to be at variance with results obtained from pendu
lum experiments. This induced him to alter the very foun
dation of his kinetic theory of gases by assuming between
the molecules a repelling force varying inversely as the fifth
power of their distances. The founders of the kinetic theory
had assumed the molecules of a gas to be hard elastic spheres;
but Maxwell, in his second presentation of the theory in 1866,
APPLIED MATHEMATICS. 403
went on the assumption that the molecules behave like cen
tres of forces. He demonstrated anew the law of distribution
of velocities ; but the proof had a flaw in argument, pointed
out by Boltzmann, and recognised by Maxwell, who adopted
a somewhat different form of the distributive function in a
paper of 1879, intended to explain mathematically the effects
observed in Crookes radiometer. Boltzmann gave a rigorous
general proof of Maxwell s law of the distribution of velocities.
None of the fundamental assumptions in the kinetic theory
of gases leads by the laws of probability to results in very
close agreement with observation. Boltzmann tried to estab
lish kinetic theories of gases by assuming the forces between
molecules to act according to different laws from those pre
viously assumed. Clausius, Maxwell, and their predecessors
took the mutual action of molecules in collision as repulsive,
but Boltzmann assumed that they may be attractive. Ex
periment of Joule and Lord Kelvin seem to support the latter
assumption.
Among the latest researches on the kinetic theory is Lord
Kelvin s disproof of a- general theorem of Maxwell and Boltz-
mann, asserting that the average kinetic energy of two given
portions of a system must be in the ratio of the number of
degrees of freedom of those portions.
INDEX.
Abacists, 126.
Abacus, 8, 13, 63, 79, 82, 119, 122, 126,
129.
Abbatt, 334.
Abel, 347, 348; ref. to, 146, 279, 291,
312, 328, 336, 337, 350, 353, 371.
Abelian functions, 292, 312, 328, 346,
348, 349, 352, 355-357, 359.
Abelian integrals, 350, 379.
Abel s theorem, 352.
Absolute geometry, 301.
Absolutely convergent series, 335, 337,
338.
Abul Gud, 111 ; ref. to, 113.
Abul Hasan, 115.
Abul Wef a, 110 ; ref. to, 112, 113.
Achilles and tortoise, paradox of, 27.
Acoustics, 262, 270, 278, 386.
Action, least, 253, 366, 401; varying,
292, 318, 379.
Adams, 375 ; ref. to, 214.
Addition theorem of elliptic integrals,
252, 350, 396.
Adrain, 276.
j3Equipollences, 322.
Agnesi, 260.
Agrimensores, 80.
Ahmes, 10-15 ; ref. to, 17, 18, 53, 74.
Airy, 375 ; ref. to, 383.
Al Battani, 109; ref. to, 110, 125.
Albertus Magnus, 134.
Albiruni, 111; ref. to, 102, 104.
Alcuin, 119.
Alembert, D J . See D Alembert.
Alexandrian School (first) , 34-54 ; (sec
ond), 54-62.
Alfonso s tables, 127.
Algebra: Beginnings in Egypt, 15;
early Greek, 73; Diophantus, 74-77 p
Hindoo, 93-96 ; Arabic, 107, 111, 115 ;
Middle Ages, 133, 135 ; Eenaissance,
140, 142-150, 152; seventeenth cen
tury, 166, 187, 192; Lagrange, 267;
Peacock, 284; recent, 315-331; ori
gin of terms, 107, 115. See Nota
tion.
Algebraic functions, 346; integrals,
377.
Algorithm, origin of term, 106 ; Mid
dle Ages, 126, 129.
Al Haitam, 115 ; ref. to, 112.
Al Hayyami, 112 ; ref. to, 113.
Al Hazin, 112.
Al Hogendi, 111.
Al Karhi, 111, 113.
Al Kaschi, 114.
Al Kuhi, 111; ref. to, 112.
Allegret, 377.
Allman, IX., 36.
Al Madshriti, 115.
Almagest, 56-58; ref. to, 105, 109, 127,
134, 136, 140.
Al Mahani, 112.
Alphonso s tables, 127.
Al Sagani, 111.
Alternate numbers, 322.
Ampere, 394; ref. to, 361.
Amyclas, 33.
Analysis (in synthetic geometry), 30,
39; Descartes , 186; modern, 331-
334.
Analysis situs, 226, 315.
Analytic geometry, 185-189, 191, 193,
240, 287, 307-315.
405
406
INDEX.
Analytical Society (in Cambridge),
283.
Anaxagoras, 18 ; ref . to, 28.
Anaximander, 18.
Anaximenes, 18.
Angeli, 185.
Anger, 375.
Anharmonie ratio, 178, 294, 297, 306.
Anthology, Palatine, 73, 120.
Antiphon, 26 ; ref. to, 27.
Apices of Boethius, 82; ref. to, 63,
103, 119, 126, 129.
Apollonian Problem, 50, 154, 188.
Apollonius, 45-50 ; ref. to, 35, 37, 40,
54, 61, 66, 78, 105, 108, 115, 140, 153,
154.
Appel, 346.
Applied mathematics, 373-403. See
Astronomy, Mechanics.
Arabic manuscripts, 124-128.
Arabic numerals and notation, 3, 73,
87, 102, 112, 127-129, 159.
Arabs, 100-117.
Arago, XI., 332, 392.
Arbogaste, 260.
Archimedes, 40-45 ; ref. to, 2, 35, 37,
39, 45, 47, 49, 50, 54, 61, 65, 73, 78, 90,
105, 108, 140, 144, 369, 173, 182.
Archytas, 23; ref. to, 29, 31, 32, 43.
Areas, conservation of, 253.
Arenarins, 65.
Argand, 317; ref . to, 264.
Aristsens, 34; ref. to, 46.
Aristotle, 34; ref. to, 9, 17, 27, 43, 61,
68, 125.
Arithmetic: Pythagoreans, 20, 67-70 ;
Platonists, 29 ; Euclid, 38, 70 ; Greek,
63-77; Hindoo, 90-92; Arabic, 106;
Middle Ages, 119, 122, 123, 126, 130,
133, 134; Renaissance, 150, 151, 158-
161. See Numbers, Notation.
Arithmetical machine, 220, 284.
Arithmetical triangle, 196.
Armemante, 313.
Arneth, X.
Aronhold, 327.
Aryabhatta, 86 ; ref. to, 88, 91, 98,
Aschieri, 305, 306.
Assumption, tentative, 75, 92. See
Eegula falsa.
Astrology, 155.
Astronomy: Babylonian, 8; Egyptian,
10; Greek, 18, 24, 32, 39, 51, 56 ; Hin
doo, 86; Arabic, 100, 101, 105, 115;
Middle Ages, 127; Newton, 212-216;
more recent researches, 253, 257,
262, 271-274, 366, 373-3TT. See Me
chanics.
Athelard of Bath, 125 ; ref. to, 135.
Athensens, 32.
Atomic theory, 382.
Attains, 46,
Attraction, 277. See Gravitation, El
lipsoid.
August, 296.
Ausdehnungslehre, 320, 321, 378.
Axioms (of geometry), 30, 37, 38, 281,
300, 315.
Babbage, 283, 356.
Babylonians, 5-9; ref. to, 19, 51.
Bachet de Meziriac. See Meziriac.
Bachmann, 371; ref. to, 365.
Bacon, R., 134.
Baker, Th., 113.
Ball, Sir B. S., 378.
Ball, W. W. B., X., 217.
Ballistic curve, 279.
Baltzer, R., 314; ref, to, 302, 325.
Barbier, 341.
Barrow, 198; ref. to, 173, 202, 203,
221, 227.
Basset, 380, 382.
Battaglini, 306.
Bauer, XII.
Bamngart, XI.
Bayes, 340.
Beaumont, XI.
Beaune, De. See De Beanne.
Bede, the Venerable, 118.
Beer, 392.
Beha Bddin, 114.
Bellavitis, 322; ref. to, 300, 304, 317.
Beltrami, 304,305; ref. to, 315.
Ben Junus, 115.
Berkeley, 236.
Bernelinus, 122.
Bernoulli, Daniel, 238; ref. to, 255,
262, 386, 401.
Bernoulli, Nicolaus (born 1695), 238.
Bernoulli, Nicolaus (born 1687), 239,
251, 269.
INDEX.
40T
Bernoulli, James (born 1654) , 237, 238
ref . to, 182, 226, 229, 251.
Bernoulli, James (born 1758), 239, 356
387.
Bernoulli, John (born 1667), 238; ref.
to, 226, 229, 232, 234, 237, 243, 250
251, 356.
Bernoulli, John (born 1710), 239.
Bernoulli, John (born 1744), 239.
Bernoullis, genealogical table of, 236.
Bernoulli s theorem, 237.
Bertini, 305.
Bertrand, 337, 340, 342, 377, 379, 380,
401.
Bessel, 373-375; ref. to, 303, 309, 351.
Bessel s functions, 374,
Bessy, 181.
Beta function, 249.
Betti, 353.
Beyer, 160.
Bezout, 260; ref. to, 250, 264.
Bezout s method of elimination, 260
331.
Bhaskara, 87; ref. to, 92-95, 97, 152.
Bianchi, 328.
Billingsley, 138.
Binet, 324, 387.
Binomial formula, 195. 196, 202, 251
348.
Biot, 275, 288, 393.
Biquadratic equation, 112, 146, 149.
Biquadratic residues, 366.
Biquaternions, 378.
Bjerknes, C. A., XIII., 357, 383.
Bobillier, 308.
Bocher, XIV.
Bode, 341.
Boethius, 81 ; ref. to, 63, 72, 103, 118,
121, 134, 135.
Bois-Reymond, P. du, XIIL, 337-339, 362.
Boltzmann, 397, 403.
Bolyai, Johann, 302; ref. to, 291.
Bolyai, Wolfgang, 301, 302; ref. to,
291, 364.
Bolza, 350.
Bombelli, 146; ref. to, 152.
Bonnet, 0., 314; ref. to, 337, 342.
Boole, 343; ref. to, 291, 325, 340, 341,
346.
Booth, 311.
Borchardt, 355.
Bouniakowsky, 365.
Bouquet, 344; ref. to, 346, 354.
Bour, 341, 377.
Boussinesq, 383, 391, 393.
Bowditch, 275, 323.
Boyle s law, 401.
Brachistochrone (line of swiftest de
scent), 234,238.
Bradwardine, 135 ; ref. to, 141.
Brahe, Tycho, 110, 139, 168.
Brahmagupta, 86; ref. to, 92, 95, 98,
102.
Bredon, 135.
Bretschneider, IX., 97, 320.
Brianchion, 178, 288, 289.
Briggs, 163.
Brill, A., 297, 311; 356.
Brill, L., 307.
Bring, 328.
Brioschi, 327 ; ref. to, 325, 330, 334, 350,
353, 379.
Briot, 344; ref. to, 346, 354.
Brouncker, 197.
Bruno, Faa de, 327.
Brans, 377.
Bryson of Heraclea, 27.
Buchheim, 378 ; ref. to, 306.
Buckley, 159.
Budan, 282.
Buddha, 89.
Buffon, 340.
Bungus, 155.
Biirgi, 160; ref. to, 165.
Burkhardt, H., XII., 328.
Burkhardt, J. K., 275.
Burmester, 300.
Busche, 365.
Buteo, 154.
Buy s-Ballot, 402.
Byrgius. See Biirgi.
Caesar, Julius, 81.
Calculating machines, 220, 284.
Calculation, origin of word, 79.
Calculus. See Differential Calculus.
Calculus of operations, 292; of varia
tions, 247, 249, 261, 265, 296, 328,
333-334, 356, 366.
Calendar, 9, 81, 141, 154, 271.
Callisthenes, 9.
Canon paschalis, 79.
408
INDEX.
Cantor, G., 339, 362, 372.
Cantor, M., IX., X., 112.
Capelli, 330.
Capillarity, 278, 366, 388, 398.
Caporali, 313.
Cardan, 144; ref. to, 149, 152, 155, 156,
159
Carll,* 334.
Carnot, Lazare, 288, 289; ref. to, 56,
236, 293.
Carnot, Sadi, 399.
Casey, 313.
Cassini, D., 257.
Cassiodorius, 83, 118.
Casting out the 9 s, 91, 106.
Catalan, E., 325.
Cataldi, 159.
Catenary, 191, 234, 237.
Cattle-problem, 73.
Cauchy, 331-333; ref. to, 243, 247, 264,
322, 324, 328, 330, 335, 338, 339, 341,
342, 345, 348, 349, 350, 353, 356, 362,
368, 383, 387, 388, 390, 393.
Caustics, 238, 241.
Cavalieri, 170; ref. to, 167, 193, 221.
Cavendish, 394.
Cayley, 325, 326; ref. to, XII., XIV.,
291, 296, 297, 306, 308, 311, 313, 319,
324, 330, 346, 354, 355.
Centre of gravity, 177, 191 ; of oscilla
tion, 191, 243.
Centres of osculation, 49.
Centrifugal force, 183, 192, 214.
Ceulen, van. See Ludolph.
Ceva, 290.
Chapman, 324.
Characteristics, method of, 297.
Chasles, 296-298; ref. to, X., 39, 47
49, 52, 172, 289, 294, 306, 311, 313, 377
Chauvenet, 377.
Chess, 92.
Cheyne, 206.
Chinese, 19.
Chladni s figures, 386.
Chree, 382, 391.
ChristotM, 325, 327.
Circle, 19, 24-28, 31, 41, 52, 154, 194
degrees of, 7, 271 ; division of, 329
"~
, <
/Circle-sguareri 2, W, 1
Cissoid,50,
lairaut, 256-258; ref. to, 244, 252, 255,
262.
lapeyron, 399.
,larke, 341.
Jlausius, 399; ref. to, 390, 398, 400-
402.
Ulavius, 155 ; ref. to, 154.
)lebsch, 312, 313 ; ref. to, XII., 296, 309,
315, 322, 327, 328, 333, 341, 342, 358,
381, 383, 390-392.
Jlifford, 305, 306; ref. to, 297, 319, 324,
358, 378, 396.
Cockle, 315.
Colburn, Z., 180.
folding, 398.
Cole, 330.
Colebrooke, 87.
Colla, 143, 145.
Jollins, 203, 223, 227, 228, 230, 232.
olson, 204.
Combinatorial School, 247, 335.
Commandinus, 153.
Commercium epistolicum, 206, 232.
Jomplex quantities, 292, 317. See Im-
aginaries.
Complex of lines, 309.
Computus, 118, 119.
"Jomte, X.
Concentric spheres of Eudoxus, 32.
Conchoid, 50.
Condensation of singularities, 362.
Conform representation of surfaces,
361.
Congruencies, theory of, 365.
Congruency of lines, 309. t ^
Conic sections, Greek, J^J4, 4,jtf,
45-49, 55; Arabs, 101, 112; Renais
sance, 153; Kepler, 168; more re-,
cent researches, 176-178, 192. See
Geometry.
Conon,40; ref. to, 42.
Conservation of areas, 253 ; of energy,
397, 398 ; of vis viva, 192.
Continued fractions, 159, 197, 252, 270.
Continuity, 169, 193, 226, 293, 333, 359,
372.
Contracted vein, 384.
Contravariants, 326.
Convergence of series, 334-339.
Co-ordinates, 185, 294, 308, 314, 379;
first use of term, 226.
INDEX.
409
Copernican System, 139.
Copernicus, 56, 139.
Correspondence, principle of, 293, 297.
Cosine, 165.
Coss, term for algebra, 152.
Cotangent, 141, 165.
Cotes, 242 ; ref . to, 243.
Coulomb, 394.
Cournot, 340.
Cousinery, 299.
Covariants, 327, 354, 369.
Cox, 306.
Craig, 306, 347, 355, 382.
Craige, 226.
Cramer, 217.
Crelle, 348; ref . to, 349.
Crelle s Journal, 295.
Cremona, 299; ref. to, 291, 294-296,
300, 313.
Cridhara, 87.
Criteria of convergence, 334-339.
Crofton, 341.
Crozet, 288.
Ctesibius, 52.
Cube, duplication of. See Duplication
of the cube.
Cube numbers, 72, 111, 180.
Cubic curves, 217, 257, 297.
Cubic equations, 112, 113, 142-145, 149,
152, 153. See Algebra.
Cubic residues, 366.
Culmann, 299, 300.
Curtze, M., 299.
Curvature, measure of, 314.
Curve of swiftest descent, 234, 238.
Curves, osculating, 226 ; quadrature of,
42, 49, 177, 190, 193, 202, 220; theory
of, 226, 240, 242, 243, 292, 321. See
Cubic curves, Rectification, Geom
etry, Conic sections.
Cusanus, 154.
Cyclic method, 96, 97.
Cycloid, 171, 173, 176, 187, 190, 191,
225, 234, 240.
Cyzicenus, 33.
Czuber, 340.
D Alembert, 254-256 j ref. to, 254, 258,
262, 265, 268-270, 386.
D Alembert s principle, 254.
Darnascius, 61 ; ref. to, 38, 104.
! Darbous:, XIII., 313, 343, 346, 361, 362.
I Darwin, 376; ref. to, 384, 391.
Data (Euclid s), 39.
Davis, E. W., 306.
Davis, "W. M., 385.
De Baune, 189; ref. to, 185, 223, 225.
Dee, 138.
Decimal fractions, 159-161.
Decimal point, 161.
Dedekind, 371 ; ref. to, 357, 362, 372.
Deficiency of curves, 312.
Definite integrals, 109, 334, 339, 341,
351, 362.
Deinostratus. See Dinostratus.
De Lahire, 285, 290.
Delambre, 366.
Delaunay, 376 ; ref. to, 333, 334.
Delian problem. See Duplication--^
the cube.
Del Pezzo, 305.
Democritus, 28 ; ref. to, 16.
De Moivre, 240.
De Morgan, 316 ; ref. to, X., XL, 1, 2,
70, 96, 161, 205, 229, 233, 260, 277,
285, 291, 333, 337, 340, 356.
De Paolis, 306.
Derivatives, method of, 269.
Desargues, 177; ref. to, 174, 184, 240,
285, 290.
Desboves, 379.
Descartes, 183-189; ref. to, 4, 48, 60,
113, 167, 173, 174, 189, 191, 192, 216,
220, 223, 240, 317 ; rule of signs, 187,
193.
Descriptive geometry, 286-288, 300.
Determinants, 226, 265, 278, 313, 324,
325, 334, 363.
Devanagari-numerals, 103.
Dialytic method of elimination, 330.
Differences, finite. See Finite dif
ferences.
Differential calculus, 200,221-227, 236-
242 (see Bernoullis, Euler, La-
grange, Laplace, etc.) ; controversy
between Newton and Leibniz, 227-
233; alleged invention by Pascal,
174; philosophy of ,-236, 256, 259, 268,
289, 333.
Differential equations, 239, 252, 265,
278, 314, 318, 321, 333, 341-347.
Differential invariants, 327.
410
INDEX.
Dingeldey, 315.
Dini, 337; ref. to, 362.
Dinostratus, 32; ref. to, 25.
Diocles, 50.
Diodorus, 10, 40, 58.
Diogenes Laertius, 17, 32.
Dionysodorus, 54.
Diophantus, 74^77 ; ref. to, 55, 61, 86,
93, 95, 96, 105, 106, 107, 110, 111, 179,
372.
Directrix, 49, 60.
Dirichlet, 367-369; ref. to, XHL, 179,
291, 334, 338, 339, 348, 356, 357, 359,
362, 371, 396.
Dissipation of energy, 400. .
Divergent parabolas, 217, 257.
Divergent series, 255, 337.
Division of the circle, 7, 271, 329,
365.
Diwani-numerals, 102.
Donkin, 379.
Dositheus, 40.
Dostor, 325.
Dove, 384.
D Ovidio, 306.
Dronke, XII.
Duality, 290, 297, 308.
Dubamel, 333, 388.
Diihring, E., X.
Duillier, 230.
Duodecimals, 124, 126.
Dupin, 288, 289 ; ref. to, 300,., 314. .. .
Duplication of tne cube, 23-25, 31, 32,
45,50,153. -~-"~- ~ " /
Durege, 354; ref. to, 309, 315.
Diirer, A., 156.
Dusing, 340.
Dyck, 315. See Groups.
Dynamics, 318, 378-381.
Dziobek, XIII. , 377.
Earnshaw, 383.
Earth, figure of, 257, 292; rigidity of
391; size of , 214, 215.
Eddy, 300.
Edfu, 12, 53.
Edgeworth, 340.
Egyptians, 9-16, 19.
Eisenlohr, 333.
Eisenstein, 369; ref. to, 354, 357, 365
370, 371.
ilastic curve, 237.
llasticity, 278, 387-392.
Electricity, 394-398.
Electro-magnetic theory of light, 394.
Elements (Euclid s), 36-39, 104, 114,
125, 127, 128, 133, 135, 136, 138. See
Euclid.
Elimination, 250, 308, 310, 330, 331. See
Equations.
Elizabeth, Princess, 188.
Ellipsoid (attraction of), 215, 277, 280,
285, 298, 366, 378, 379; motion of,
383.
Elliptic co-ordinates, 379.
Elliptic functions, 241, 279, 280, 296,
329, 345, 346, 347-354, 363, 367, 370.
Elliptic geometry. See Non-Euclidean
geometry.
Elliptic integrals, 247, 252, 328, 349,
350.
Sly, 372. *
Encke, 366.
Energy, conservation of, 397, 398.
Enestrom, XI.
Inneper, 353 ; ref. to, XIII.
Entropy, 400.
Enumerative geometry, 297.
Epicycles, 51.
Epping, IX., 9.
Equations, solution of, 15, 149, 153,
186, 250, 260, 263, 277, 348; theory
of, 75, 166, 189, 193, 216, 240, 241,
250, 328-331; numerical, 147, 264,
282. See Cubic equations, Algebra,
Theory of numbers.
Eratosthenes, 44; ref. to, 25, 35, 40, 71.
Errors, theory of. See Least squares.
Espy, 384.
Ether, luminiferous, 393.
Euclid, 35-40, 70, 71; ref. to, 17, 21,
22, 26, 30, 31, 33, 34, 42, 46, 50, 53,
57, 58, 61,72, 73, 78, 81, 97, 104, 108,
114, 125, 127, 136, 138, 144, 162, 281,
303.
Euclidean space. See Non-Euclidean
geometry.
Eudemian Summary, 17, 21, 30, 32, 33,
35.
Eudemus, 17, 22, 45, 46, 69.
Eudoxus, 32, 33; ref. to, 16, 28, 31, 32,
35, 36, 51.
ISTDEX.
411
Euler, 248-254; ref. to, 77,96, 179, ISO,
239, 241, 246, 250, 258, 259, 261, 262,
264, 265, 267, 268, 273, 278, 279, 280,
287, 314, 317, 334, 364, 365, 367, 374,
386, 387, 401.
Eulerian integrals, 280.
Eutocius, 61 ; ref. to, 45, 46, 54, 65.
Evolutes, 49, 191.
Exhaustion, method of, 26, 28, 33, 36,
42, 169.
Exponents, 134, 152, 160, 162, 187, 202,
241.
Factor-tables, 368.
Fagnano, 241.
Fahri des Al Karhi, 111.
Falsa positio, 92, 147.
Faraday, 398.
Favaro, XII.
Faye, 377.
Fermat, 173, 179-182; ref. to, 172, 173,
177, 198*252, 264, 265, 367.
Fermat s theorem, 180, 252.
Ferrari, 145 ; ref. to, 144, 264.
Ferrel, 384; ref. to, 376.
Ferro, Scipio, 142.
Fibonacci. See Leonardo of Pisa.
Fiedler, 300, 312, 328.
Figure of the earth, 257, 292.
Finseus, 159.
Fine, XII.
Finger-reckoning, 63, 118.
Finite differences, 240, 242, 251, 270,
278, 343.
Fink, XII.
Fitzgerald, 394.
Flachenabbildung, 313.
Flamsteed, 218.
Floridas, 142, 144.
Flexure, theory of, 389.
Fluents, 205, 206.
Fluxional controversy, 227-233.
Fluxions, 200, 202-213, 333.
Focus, 49, 60, 169, 170.
Fontaine, 252, 254.
Forbes, 397.
Force-function, 395. See Potential.
Forsyth, XII., 327, 344, 362.
Four-point problem, 341.
Fourier, 281-284; ref. to, 174, 255, 351,
356, 367.
Fourier s series, 283, 338, 339, 367,
386.
Fourier s theorem, 282.
Fractions, Babylonian, 7; Egyptian,
13; Greek, 26, 64, 65; Eoman, 78;
Hindoo, 94; Middle Ages, 120, 124;
decimal, 159, 160; sexagesimal, 7,
57, 65, 67, 126; duodecimal, 124, 126;
continued, 159, 197, 252, 270. See
Arithmetic.
Franklin, 327, 372.
Frantz, XIII.
Fresnel, 392.
Fresnel s wave-surface, 209, 314.
Frezier, 286.
Fricke, 354.
Friction, theory of, 382.
Frobenius, 325, 344, 345.
Frost, 315.
Froude, 381, 384.
Fuchs, 343; ref. to, 344, 345.
Fuchsian functions, 345, 360.
Fuchsian groups, 345.
Functions, definition of, 356; theory
of, 268, 269, 345, 356-362; arbitrary,
262, 283. See Elliptic functions,
Abelian functions, Hyperelliptic
functions, Theta functions, Beta
function, Gamma function, Omega
function, Sigma function, ^ Bessel s
function, Potential.
Funicular polygons, 299.
Gabir ben Aflah, 115; ref. to, 127.
Galileo, 182; ref. to, 43, 139, 161, 168,
170, 171, 188.
Galois, 329.
Gamma function, 249.
Garbieri, 324.
Gases, Kinetic theory of, 401-403.
Gauss, 363-367; ref. to, 77, 158, 247,
248, 251, 264, 276, 291, 294, 302, 303,
304, 313, 314, 315, 317, 320, 324, 325,
330, 333, 335, 343, 348, 350, 351,
357, 362, 373, 398.
Gauss Analogies, 366.
Geber. See Gabir ben Aflah.
Geber s theorem, 116.
Geminus, 53 ; ref. to, 46, 50, 57.
Gellibrand, 165.
Genocchi, 365.
412
INDEX.
Geodesies, 249, 379.
Geodesy, 366.
Geometry, Babylonian, 8; Egyptian,
10-13; Greek, 17-62, 69; Hindoo, 97,
98; Roman, 80; Arabic, 104, 108,
110, 113, 114; Middle Ages, 121, 125,
127, 128, 130, 131; Renaissance, 138,
153, 154, 158, 1G7; analytic, 186-189,
191, 193, 287, 307-315; modern syn
thetic, 240, 285-290, 293-307; de
scriptive, 286-288, 300. See Curves,
Surfaces, Curvature, Quadrature,
Rectification, Circle.
Gerard of Cremona, 126.
Gerbert, 120-124.
Gergonne, 297 ; ref . to, 178, 290.
Gerhardt, XL, 227, 230, 233.
Gerling, 366.
Germain, Sophie, 387 ; ref. to, 386.
German Magnetic Union, 366.
Gerstner, 389.
Gibbs, 400; ref. to, XII., 319.
Giovanni Campano, 127.
Girard, 166; ref. to, 127, 161.
Glaisher, 372; ref. to, XIII., 325, 328,
374, 368.
Glazebrook, 397; ref. to, XIV.
Gobar numerals, 82, 103.
Godfrey, 218.
Golden section, 33.
Gopel, 355.
Gordan, 312, 327, 330.
Gournerie, 300, 311.
Goursat, 343 ; ref. to, 350.
Gow, IX., 35.
Graham, XII.
Grammateus, 151.
Grandi, 251.
Graphical statics, 292, 299.
Grassmann, 320-321 ; ref. to, 294, 304,
317, 318, 378.
Gravitation, theory of, 213, 258, 271,
275.
Greeks, 16-77.
Green, 395; ref. to, 358, 383, 388, 390,
393, 395.
Greenhill, 354, 382.
Gregorian Calendar, 154.
Gregory, David F., 215, 284, 315.
Gregory, James, 228, 243.
Gromatici, 80.
Groups, theory of, 328-330, 34^-346.
Papers by W. Dyck (Math. Ann., 20
and 22) and by O. Holder (Math.
Ann., 34) should have been men
tioned on p. 330.
Grunert, 314 ; ref. to, 320.
Gua, de, 240.
Gubar-numerals, 82, 103.
Guderniann, 353.
Guldm, 167; ref. to, 59, 171.
Guldinus. See Guldin.
Gunter, E., 165.
Giinther, S., IX,, X., XI., 325.
Gutzlaff, 353.
Haan, 334.
Haas, XII.
Hachette, 288, 300.
Hadamard, 368.
Hadley, 218.
Hagen, 276.
Halifax, 134; ref. to, 136.
Halley, 45, 213, 214, 261.
Halley s Comet, 258, 374.
Halphen, 311; ref. to, 297, 315, 327,
344, 345, 354.
Halsted, X., 303.
Hamilton, W., 184, 316.
Hamilton, W. R., 318, 319; ref. to,
266,291, 292, 314, 31(5, 317, 321, 324,
328, 341, 378, 379, 393, 401.
Hamilton s numbers, 329.
Hammond, J., 327.
Hankel, 322; ref. to, IX,, X. ? 28, 93,
96, 285, 325, 339, 362.
Hann, 385.
Hansen, 375.
Hanus, 325.
Hardy, 174.
Harkness, 362.
Harmonics, 55.
Saroun-al-Raschid, 104.
Harrington, 377.
Harriot, 166; ref. to, 147, 152, 162, 187,
192.
Hathaway, XI.
Heat, theory of, 399-401.
Heath, 306:
fieaviside, 319, 397, 398.
Hebrews, 19.
Hegel, 373.
INDEX.
413
Heine, 339; ref. to, 362, 372.
Helen of geometers, 187.
Helicon, 32.
Heliotrope, 363.
Helmholtz, 397, 398; ref. to, 304, 305,
382, 385, 386, 393, 396, 400, 401.
Henrici, XIII.
Henry, 396.
Heraclid.es, 45.
Hermite, 353; ref. to, XIII., 328, 330,
343, 347, 350, 355, 362, 372.
Hermotimus, 33.
Herodianic signs, 63.
Heron the Elder, 52 ; ref. to, 50, 54, 65,
80, 98, 105, 131, 140.
Herschel, J. F. W., 386; ref. to, X.,
276, 283, 284, 356.
Hesse, 309-311; ref. to, 295, 309, 312,
325, 329, 330, 333, 342, 377.
Hessian, 295, 310, 327.
Heuraet, 190.
Hexagrammum mysticum, 178, 296.
Hicks, 382, 384.
Hilbert, 327.
Hill, 376.
Hindoos, 84-100 ; ref. to, 3.
Hipparclras, 51: ref. to, 54, 56.
Hippasus, 22.
Hippias of Elis, 25.
Hippocrates of CMos, 25, 28, 30.
Hippopede, 51.
Him, 400.
History of mathematics, its value, 1-4.
HodgMnson, 389.
Holder, 0. See Groups.
Holmboe, 336, 347, 350.
Homogeneity, 293, 308.
Homological figures, 178.
Honein ben Ishak, 104.
Hooke, 213.
Hoppe, 306.
Homer, 147, 330.
Hospital, P, 239, 240.
Houel, 319.
Hovarezmi, 106; ref. to, 107, 110, 114,
125, 127.
Hudde, 189; ref. to, 203.
Hurwitz, 358.
Hussey, 377.
Huygens, 190-192; ref. to, 177, 182,
188, 213, 214, 219, 234, 257, 392.
Hyde, 321.
Hydrodynamics, 239, 255, 380, 381-384.
See Mechanics.
Hydrostatics, 44, 255. See Mechanics.
Hypatia, 61 ; ref. to, 37.
Hyperbolic geometry. See Non-Eucli
dean geometry.
Hyperelliptie functions, 292, 328, 348,
354, 360.
Hyperelliptie integrals, 352.
Hypergeonaetric series, 335, 361.
Hyperspace, 304, 305.
Hypsicles, 51 ; ref. to, 7, 38, 71, 104.
lamblichus, 72; ref. to, 10, 22, 69.
Ibbetson, 392.
Ideal numbers, 371.
Ideler, 32.
lehuda ben Mose Cohen, 127.
Ignoration of co-ordinates, 380.
Images, theory of, 381.
Imaginary geometry, 301.
Imaginary points, lines, etc., 298.
Imaginary quantities, 146, 166, 241,
287, 349, 363, 372.
Imschenetzky, 342.
Incommensurables, 36, 38, 70. See
Irrationals.
Indeterminate analysis, 95, 101, 111.
See Theory of numbers.
Indeterminate coefficients, 186.
Indeterminate equations, 95, 101, 111.
See Theory of numbers.
Indian mathematics. See Hindoos.
Indian numerals. See Arabic numer
als.
Indices. See Exponents.
Indivisibles, 170-173, 176, 193.
Induction, 340.
Infinite products, 349, 354.
Infinite series, 197, 203, 208, 220, 247,
250, 255, 259, 269, 283, 334r-339, 348,
349, 361, 363.
Infinitesimal calculus. See Differen
tial calculus.
Infinitesimals, 135, 169, 207, 208, 211.
Infinity, 27, 135, 169, 178, 193, 269, 293,
304, 308 ; symbol for, 193.
Insurance, 239, 340.
Integral calculus, 171, 223, 348, 350,
368, 371 ; origin of term, 237.
414
INDEX.
Interpolation, 194.
Invariant, 293, 310, 325, 328, 344, 354,
Inverse probability, 340.
Inverse tangents (problem of), 169, 189,
220, 223, 223.
Involution of points, 60, 177.
Ionic School, 17-19.
Irrationals, 22, 26, 69, 94, 107, 362, 372.
See Incommensurables.
Irregular integrals, 344.
Ishak ben Honein, 104.
Isidores of Seville, 118; ref. to, 61.
Isochronous curve, 234.
Isoperimetrical figures, 51, 237, 249,
261. See Calculus of variations.
Ivory, 285; ref. to, 276.
Ivory s theorem, 285.
Jacohi, 351-352; ref. to, 279, 291, 295,
308, 309, 315, 324, 330, 333, 341, 347,
349, 360, 353, 357, 365, 367, 370, 374,
377, 378, 379, 381.
Jellet, 334; ref . to, 381, 390.
Jerrard, 328.
Jets, 382, 386.
Jevons, 340.
Joachim. See Rhseticus.
Jochmann, 402.
John of Seville, 126, 159.
Johnson, 347.
Jordan, 329; ref. to, 341, 343, 346.
Jordanus Nemorarius, 134.
Joubert, 353.
Joule, 399; ref. to, 401, 403.
Julian calendar, 81.
Jurin, 236.
Kaestner, 363; ref. to, 217.
Kant, 274, 376. .
Kautimann. See Mercator, N.
Kaffl, 231, 232, 235.
Kelland, 383, 397.
Kelvin, Lord, 395-396; ref. to ,283, 315
358, 381, 382, 388, 388, 391, 393, 394
395, 399, 400, 403. See Thomson, \V
Kempe, 326.
Kepler, 168-170; ref. to, 139, 156, 158
161, 167, 171, 174, 202, 213, 263.
Kepler s laws, 168,213.
Kerbedz, XIII.
Ketteler, 393.
Killing, 306.
inckhuysen, 204.
Kinetic theory of gases, 401-403.
Sirchhoff, 396; ref. to, 309, 381, 382,
383, 388, 393, 396.
llein, 343; ref. to, XII., 305, 306, 307,
309, 313, 328, 330, 345, 346, 347, 354.
iGeinian groups, 345.
Heinian functions, 360.
BLohlrausch, 394.
ohn, 337.
Konig, 401.
Konigsberger, 353; ref. to, 344, 350,
354, 355.
Kopcke, 382.
lorkine, 372 ; ref. to, 341.
Korndorfer, 313.
Kowalevsky, 380 ; ref. to, 345, 352, 378.
rause, 355.
Krazer, 355.
Kronecker, 329 ; ref. to, 328, 330, 359, 365.
Kronig, 401.
Kuhn, H., 317.
Kuhn, J., 219.
Kuramer, 370, 371; ref. to, XIII., 179,
314, 337, 338, 343, 355, 365.
Lacroix, 284, 286, 287, 320.
Laertius, 10.
Lagrange, 260-270; ref. to, 4, 77, 174,
179, 183, 2*44, 246, 247, 248, 254, 255,
259, 273, 277, 278, 279, 280, 293, 296,
304, 309, 313, 314, 325, 362, 363, 364,
367, 378, 383, 386, 387, 401.
Laguerre, 306,
Lahire, de, 240.
Laisant, 319.
La Louere, 177.
Lamb, 378, 382, 383, 396.
Lambert, 258-259; ref. to, 2, 290, 303,
313
Lame, 389; ref. to, 367, 389, 392.
Lame s functions, 389.
Landen, 259; ref. to, 268, 279.
Laplace, 270-278; ref. to, 174, 215, 245,
246, 256, 263, 279, 285, 320, 336, 340,
362, 363, 373, 375, 376, 384, 386, 392,
395, 398.
Laplace s coefficients, 277.
Latitude, periodic changes in, 392.
Latus rectum, 48.
INDEX.
415
Laws of Laplace, 273.
Laws of motion, 183, 188, 213.
Least action, 253, 266, 401.
Least squares, 276, 281, 285, 363.
Lebesgue, 325, 333, 365.
Legendre, 278-281; ref. to, 247, 252,
259, 266, 276, 301, 349, 350, 351, 353,
365, 367.
Legendre s function, 280.
Leibniz, 219-235; ref. to, 4, 158, 176,
200, 208, 209, 210, 237, 241, 250, 251,
252, 268, 315, 334, 356.
Lemoine, 341.
Lemonnier, 267.
Leodamas, 33.
Leon, 33.
Leonardo of Pisa, 128 ; ref. to, 133, 137.
Leslie, X.
Le Verrier, 375 ; ref. to, 376.
Levy, 300, 391.
Lewis, 382.
Lexis, 340.
Leyden jar, 396.
L Hospital, 239, 240; ref. to, 229, 234.
Lie, 346; ref. to, 341, 350.
Light, theory of, 218, 390.
Limits, method of, 212, 268.
Lindelof , 334.
Lindemann, 315 ; ref. to, 2, 306, 356.
Linear associative algebra, 323.
Lintearia, 237.
Liouville, 369; ref. to, 314, 356, 365,
370, 379.
Lipschitz, 306; ref. to, 338, 375, 382.
Listing, 315.
Lloyd, 393.
Lobatchewsky, 301 ; ref. to, 291, 303.
Local probability, 340.
Logarithmic criteria of convergence,
337.
Logarithmic series, 197.
Logarithms, 158, 161-165, 168, 197, 242,
250. ~^ T
Logic, 37, 316, 323, 343.
Lommel, 375, 393.
Long wave, 383.
Loomis, 384.
Lorenz, 393.
Loria, XL
Loud, 298.
Lucas de Burgo. See Pacioli.
Rudolph, 154.
/udolph s number, 154.
jUne, squaring of, 25.
.iiroth, 358; ref . to, 362.
MacCullagh, 311 ; ref. to, 393.
Macfarlane, 319.
Machine, arithmetical, 220, 284.
\Iaclaurin, 243; ref. to, 236, 2M, 280,
285, 290.
Macmahon, 327.
Magic squares, 92, 135, 241.
Magister matheseos, 136.
Main, 377. ***?
Mainardi, 334.
Malfatti, 296, 328.
Malfatti s problem, 296, 312.
Mansion, 341.
Marie, Abbe, 279.
Marie, C.F.M., 298.
Marie, M., X., 52, 172.
Mathieu, 391 ; ref. to, 353, 377, 392,
398.
Matrices, 321, 324.
Matthiessen, X.
Maudith, 135 ; ref. to, 141.
Maupertius, 253, 257, 401.
Manrolycus, 153 ; ref. to, 155.
Maxima and minima, 49, 174, 186, 189,
208, 244, 333, 334, 339, 342.
Maxwell, 397; ref. to, 300, 376, 382,
390, 394, 396, 398, 400, 401, 402, 403.
Mayer, 398; ref . to, 376.
McClintock, 328.
McColl, 341.
McCowan, 383.
McCullagh, 311, 393.
McMahon, 328.
Mechanics: Greek, 23, 34, 43 ; Stevin
and Galileo, 158, 182; Descartes,
Wallis Wren, Huygens, Newton,
188, 191, 192, 212-216; Leibniz, 227;
Bernoullis, 237, 238; Taylor, 243;
Euler, 253; Lagrange, 266; La
place, 274; more recent work, 290,
328, 346, 377-381, 401. See Dynam
ics, Hydrodynamics, Hydrostatics,
Graphic statics, Laws of motion, As
tronomy, D J Alembert s principle.
Meissel, 353.
Mensechmus, 32; ref. to, 31, 34, 46, 113.
416
INDEX.
Menelaus, 55 ; ref . to, 57, 157. j
Mercator, G., 313. I
Mercator, N., 197 ; ref. to, 220.
Mere, 182.
Mersenne, 180, 191.
Mertens, 336, 367.
Meteorology, 384r-386.
Method of characteristics, 297.
Method of exhaustion, 28; ref. to, 33,
36, 42, 169.
Metius, 154.
Meunier, 314.
Meyer, A., 340, 341.
Meyer, G. F., 334.
Meyer, 0. E., 382, 392, 402.
Meziriac, 179; ref. to, 265.
Michelson, 394.
Middle Ages, 117-137.
Midorge, 174.
MincMn, 381.
Minding, 314.
Minkowsky, 370.
Mittag-Leffler, 359.
Mobius, 294; ref. to, 293, 320, 321, 366,
375, 377.
Modern Europe, 138 et seq.
Modular equations, 329, 353.
Modular functions, 354.
Mohammed ben Musa Hovarezmi, 106 ;
ref. to, 107, 110, 114, 125, 127.
Mohr, 300.
Moigno, 334.
Moivre, de, 245.
Mollweide, 366.
Moments in fLuxionary calculus, 205,
206.
Monge, 286-288; ref. to, 248, 259, 282,
293, 300, 314, 341.
Montmort, de, 240.
. Montucla, X., 172.
Moon. See Astronomy.
Moore, 330.
Moors, 115, 116, 125.
Moral expectation, 239.
Morley, 362.
Moschopulus, 135.
Motion, laws of, 183, 188, 213.
Mouton, 219.
Muir, XII., 325.
Miiller, X.
MuTLer, J. See Regiomontanus.
Multi-constancy, 390, 391.
Multiplication of series, 335, 336.
Vusa ben Sakir, 108.
Musical proportion, 8.
Mydorge, 177.
Nachreiner, 325.
Nagelbach, 324.
Napier, J., 162, 163; ref. to, 156, 161,
164, 165.
Napier, M., X.
Napier s rule of circular parts, 165.
Nasir Eddin, 114.
Nautical almanac, United States, 376.
Navier, 387 ; ref. to, 382, 390.
Nebular hypothesis, 274.
Negative quantities, 93, 152, 187, 256,
372. See Algebra.
Negative roots, 93, 112, 146, 149, 152,
166. See Algebra.
Neil, 190 ; ref. to, 198.
Neocleides, 33.
Neptune, discovery of, 375.
Nesselmann, 76.
Netto, 330.
Neumann, C., 375 ; ref. to, 309, 315, 393.
Neumann, F. E., 398; ref. to, 309, 312,
390, 392, 396, 401.
Newcomb, 376 ; ref. to, 306, 307, 392.
Newton, 201-218; ref. to, 4, 50, 60,
147, 173, 186, 191, 192, 195, 200, 238,
243, 244, 252, 254, 257, 258, 262, 268,
282, 285, 290, 297, 302; 317, 330, 334,
372, 380, 386.
Newton, controversy with Leibniz,
227-233.
Newton s discovery of binomial the
orem, 195, 196.
Newton s discovery of universal grav
itation, 213.
Newton s parallelogram, 217.
Newton s Principia, 191, 208, 212-215,
229, 233, 242.
Nicolai, 366.
Nicole, 240.
Nicolo of Brescia. See Tartaglia.
Nicomachus, 72 ; ref. to, 58, 81.
Nicomedes, 50.
Nieuwentyt, 235.
Nines, casting out the, 106.
Niven, 396.
HSDEX.
417
Nolan, 376.
Non-Euclidean geometry, 38, 300-307.
Nonius, 153 ; ref . to, 154.
Notation: in algebra, 15, 75, 93, 133,
134, 149, 150, 151, 160, 167; Baby
lonian numbers, 5-7 ; Egyptian num
bers, 13; Greek numbers, 64 ; Arabic
notation, 3, 73, 87, 102, 112, 127-129,
159 ; Koman, 78 ; decimal fractions,
160; trigonometry, 249; differential
calculus, 205, 221, 222, 260, 269, 283.
See Exponents, Algebra.
Neither, 311, 313, 330, 356.
Numbers : amicable, 68, 108, 115 ; ex
cessive, 68; heteromecic, 68; per
fect, 68; defective, 68; triangular,
180; definitions of numbers, 372 ; the
ory of numbers, 55, 76, 95, 108, 119,
131, 178-182, 252, 264, 280, 362-372.
Numbers of Bernoulli, 238.
Numerals: Egyptian, 13; Babylonian,
5-7; Greek, 64; Arabic, 87, 102, 103,
112. See Apices.
Oberbeck, 386.
OEnopides, 19; ref. to, 16.
Ohm, M., 317.
Ohrtmann, X.
Gibers, 364, 373.
Oldenburg, 228.
Olivier, 300.
Omega-function, 353.
Operations, calculus of, 292.
Oppolzer, 377.
Optics, 39.
Oresme, 134 ; ref. to, 160.
Orontius, 154.
Oscillation, centre of, 191, 243.
Ostrogradsky, 333, 379.
Otho, 142.
Oughtred, 167; ref. to, 147, 161, 202.
Ovals of Descartes, 187.
?r: values for; Babylonian and He
brew, 8; Egyptian, 11; Archime
dean, 41; Hindoo, 98; Arabic, 108;
Ludolph s, 154; Wallis 5 , 194, 195;
Brouncker s, 197; Fagnano s, 241;
Leibniz s, 220 ; selection of letter TT,
250; proved to be irrational, 259,
281 ; proved to be transcendental, 2
Pacioli, 135; ref. to, 134, 142, 152, 155,
158, 196. -
Padmanabha, 87.
Palatine anthology, 73, 120.
Pappus, 58-61 ; ref. to, 35, 39, 45, 49,
50, 55, 65, 66, 153, 178, 186.
Parabola, 42, 70, 198; semi-cubical,
190. See Geometry.
Parabolic geometry. See Non-Euclid
ean geometry.
Parallelogram of forces, 183.
Parallels, 38, 281, 300, 301, 303, 306.
Parameter, 48.
Partial differential equations, 208, 255,
287, 341 et seq,, 379.
Partition of numbers, 372.
Pascal, 175-177; ref. to, 178, 182, 196,
220, 240, 284, 285, 290, 310.
Pascal s theorem, 178.
Peacock, 284; ref. to, X., 130, 133, 161,
283, 315.
Pearson, 391.
Peaucellier, 326.
Peirce, B., 323; ref. to, 291, 317, 376,
381,
Peirce, C. S., 323; ref. to, 37, 307, 321.
Peletarius, 166.
Pell, 147, 151, 181, 219.
Pell s problem, 97, 181.
Pemberton, 201.
Pendulum, 191.
Pepin, 365.
Perier, Madame, X.
Periodicity of functions, 349, 350.
Pemter, J. M., 385.
Perseus, 50.
Perspective, 177. See Geometry.
Perturbations, 273.
Petersen, 365.
Pfaff, 341, 342; ref. to, 362.
Pfaffian problem, 341, 342.
Pherecydes, 20.
Philippus, 33.
Philolaus, 22; ref. to, 28, 68.
Philonides, 46.
Physics, mathematical. See Applied
mathematics.
Piazzi, 373.
Picard, E., 347, 350, 360.
Picard, J., 214, 215.
Piddington, 384.
418
INDEX.
Piola, 388.
Pitiscus, 142.
Plana, 375, 387, 396.
Planudes, M. ? 135.
Plateau, 382.
Plato, 29-31 ; ref. to, 3, 10, 16, 23, 32, 33,
34, 35, 63, 68.
Plato of Tivoli, 109, 125.
Plato Tiburtinus. See Plato of Tivoli.
Platonic figures, 39.
Platonic School, 29-34.
Playfair, X M 156.
Plectoidal surface, 60.
Pliicker, 307-309 ; ref. to, 304, 308, 313.
Plus and minus, signs for, 150.
Pohlfce, 300.
Poincare, 343; ref. to, XIII., 345, 346,
347, 352, 359, 368, 384, 400.
Poinsot, 378; ref. to, 377.
Poisson, 387; ref. to, 175, 298, 330, 333,
351, 375, 378, 382, 383, 386, 387, 390,
392, 395, 396.
Poncelet, 289, 290; ref. to, 178, 288,
293, 306, 308, 389.
Poncelet s paradox, 308.
Porisms, 39.
Porphyrius, 55.
Potential, 277, 358, 395.
Poynting, 397, 398.
Preston, 400.
Primary factors, Weierstrass theory
of, 354, 360.
Prime and ultimate ratios, 198, 212,
268.
Prime numbers, 38, 45, 71, 179, 180,
368.
Princess Elizabeth, 188.
Principia (Newton s), 191, 208, 212-
215, 229, 233, 242.
Pringsbeim, 336-338.
Probability, 158, 182, 192, 237, 239,
240, 245, 252, 270, 276, 285, 340, 341.
Problem of Pappus, 60.
Problem of three bodies, 253, 256, 377.
Proclus, 61 ; ref. to, 17, 19, 33, 35, 38,
39, 50, 54, 58.
Progressions, first appearance of arith
metical and geometrical, 8.
Projective geometry, 307.
Proportion, 17, 22, 23, 26, 33, 36, 38,
67, 68.
Propositiones ad acuendos iuvenes,
119, 120.
Prym, 355.
Ptolemseus. See Ptolemy.
Ptolemaic System, 56.
Ptolemy, 56-58; ref. to, 7, 9, 54, 55,
98, 104, 106, 108, 109, 115, 139, 313.
Puiseux, 356.
Pulveriser, 95.
Purbach, 134 ; ref. to, 140.
Pythagoras, 19-23, 67-70; ref. to, 3,
16, 18, 24, 29, 36, 63, 82, 97,- 135.
Pythagorean School, 19-23.
Quadratic equations, 76, 93, 107, 111,
112. #ee Algebra, Equations.
Quadratic reciprocity, 252, 280, 365.
Quadratrix, 25, 32, 59, 60.
Quadrature of the circle. See Circle;
also see Circle-squarers, TT.
Quadrature of curves, 42, 49, 177, 190,
193, 220, 222.
Quaternions, 318, 319 ; ref. to, 317.
Quercu, a, 154.
Quetelet, 340 ; ref. to, X.
Raabe, 337.
Raclau, 377.
Radiometer, 403.
Rahn, 151.
Ramus, 153.
Rankine, 399 ; ref. to, 400.
Rari-constancy, 390.
Ratios, 372.
Rayleigh, Lord, 386 ; ref. to, 374, 383,
384, 397, 398.
Reaction polygons, 300.
Reciprocal polars, 290.
Reciprocants, 327, 361.
Recorde, 151; ref. to, 158.
Rectification of curves, 169, 177, 190,
198. See Curves.
Redfield, 384.
Eeductio ad absurdum, 28.
Reech, 381. ,
Regiomontanus, 140, 141 ; ref. to, 139,
149, 153, 154, 155, 158, 160.
Regula aurea. See Falsa positio.
Regula duorum falsormn, 106.
Regula falsa, 106.
Regular solids, 21, 31, 34, 38, 51,110, 168,
INDEX.
419
Reid, 384.
Reiff, XL
Renaissance, 139-156.
Resal, 377.
Reye, 290; ref. to, 305.
Reynolds, 383.
Rhseticus, 141 ; ref. to, 139, 142.
Rheticus. See Rhseticus.
Rhind papyrus, 10-15.
Riccati, 241 ; ref. to, 239.
Richard of TTallingford, 135.
Richelot, 353; ref. to, 309, 312.
Riemann, 357-359; ref. to, 304, 305,
312, 315, 339, 342, 354, 355, 356, 302,
368, 384, 392, 398.
Riemann s surfaces, 358; ref. to, 356.
Roberts, 313.
Roberval, 172 ; ref. to, 172, 187, 191.
Rolle, 241 ; ref. to, 236.
Roman mathematics in Occident, 117-
124.
Romans, 77-83.
Romanus, 154; ref. to, 142, 148, 154.
Romer, 199.
Rosenberger, XIV.
Rosenhain, 355 ; ref. to, 353.
Roulette, 171.
Routh, 380; ref. to, 381, 397.
Rowland, 382, 394, 397, 398.
Rudolff, 151.
Ruffini, 328.
Riihlmann, 400.
Rule of signs, 187, 193.
Rule of three, 92, 106.
Saccheri, 303.
Sachse, XIII.
Sacro Bosco. See Halifax.
Saint-Venant, 390; ref. to, 322, 382,
389, 393.
Salmon, XII., 295, 311-313, 330.
Sand-counter, 65, 90.
Sarrau, 393.
Sarrus, 333.
Saturn s rings, 192, 376.
Saurin, 240.
Savart, 387.
Scaliger, 154.
Schellbach, 296.
Schepp, 362.
Schering, 306; ref. to, 357, 365.
Schiaparelli, 32.
Schlafli, 306; ref. to, 338, 353.
Sehlegel, 322; ref. to, XIL, 306.
Schlessinger, 300.
Schlomilch, 375.
Schmidt, XIL
Schooten, van, 189 ; ref. to, 190, 202.
Schreiber, 288, 300.
Sehroter, H., 313; ref. to, 296, 353.
Schroter, J. H., 374.
Schubert, 297.
Schumacher, 366; ref. to, 348.
Schuster, XIII.
Schwarz, 361; ref. to, 297, 339. 345.
347, 3(52.
Schwarzian derivative, 361.
Scott, 325.
Screws, theory of, 378.
Secants, 142.
Sectio aurea, 33.
Section, the golden, 33.
Seeber, 371.
Segre, 305.
Seidel, 339.
Seitz, 341.
Selling, 371.
Sellmeyer, 393.
Semi-convergent series, 336.
Semi-cubical parabola, 190.
Semi-invariants, 328.
Serenus, 55.
Series, 111, 245. See Infinite series,
Trigonometric series, Divergent
series, Absolutely convergent series,
Semi-convergent series, Fouriet s
series, Uniformly convergent series.
Serret, 313 ; ref. to, 341, 342, 377, 379.
Servois, 284, 288, 290.
Sexagesimal system, 7, 57, 65, 67, 126.
Sextant, 218.
Sextus Julius Africanus, 58.
Siemens, 385,
Sigma-f unction, 354.
Signs, rule of, 187, 193.
Similitude (mechanical), 380.
Simony, 315.
Simplicius, 61.
Simpson, 249.
Simson, 290; ref. to, 37, 39.
Sine, 99, 102, 109, 116, 125, 140, 141;
origin of term, 109.
420
INDEX.
Singular solutions, 226, 265, 277.
Sluze, 189; ref. to, 222, 224.
Smith, A., 381.
Smith, H., 369, 370; ref. to, XIIL, 353,
372.
Smith,E,,242.
Sohnke, 353.
Solid of least resistance [Prin. IL, 25],
215.
Solitary wave, 383.
Somoff, 381.
Sophist School, 23-29.
Sosigenes, 81.
Sound, velocity of, 270, 278. See
Acoustics.
Speidell, 165.
Spherical Harmonics, 247.
Spherical trigonometry, 56, 115, 280,
294.
Spheroid (liquid) , 384.
Spirals, 42, 60, 237.
Spitzer, 333.
Spottiswoode, 325; ref. to, XII., 292.
Square root, 65, 94, 159.
Squaring the circle. See Quadrature
of the circle.
Stabl, 306.
Star-polygons, 22, 135, 156.
Statics, 44, 182. See Mechanics.
Statistics, 340.
Staudt, von. See Von. Staudt.
Steele, 381.
Stef ano, 382.
Steiner, 295, 296 ; ref. to, 293, 297, 298,
08, 311, 312, 320, 348, 357.
Stereometry, 31, 33, 38, 168.
Stern, 357, 365.
Stevin, 160 ; ref. to, 134, 162, 182.
Stevimis. See Stevin.
Stewart, 290.
Stifel, 151 ; ref. to, 149, 151, 155, 162.
Stirling, 244.
Stokes, 381; ref. to, 339, 3S2, 383, 386,
388, 390, 393, 398.
"Italy, 306.
Strassmaier, IX.
Strauch, 334.
Strings, vibrating, 242, 255, 262.
Stringhana, 306.
Strutt, J, W., 386. See Rayleigh.
Struve, 366.
Sturm, J. C. F., 330; ref. to, 178, 282,
379, 381.
Sturm, R., 296.
Sturm s theorem, 330.
St. Vincent, Gregory, 190, 197.
Substitutions, theory of, 292.
Surfaces, theory of, 250, 287, 295, 299,
309, 310, 314.
Suter, X.
Swedenborg, 274.
Sylow, 33JO; ref. to, 350.
Sylvester, 326; ref. to, SHI., 216, 296,
310, 311, 312, 319, 324, 325, 328, 330,
341, 361, 368, 370, 378, 396.
Sylvester II. (Gerbert), 120-124.
Symmetric functions, 250, 328, 330.
Synthetic geometry, 293-307.
Synthesis, 30, 31.
Taber, 324.
Tabit ben Korra, 108 ; ref. to, 105.
Tait, 283, 319, 381, 388, 400.
Tangents, in geometry, 62, 173, 186;
in trigonometry, 110, 141, 142.
Tangents, direct problem of, 198, 223 ;
inverse problem of, 169, 189, 220, 222,
223.
Tannery, 343 ; ref. to, 362.
Tartaglia, 143-145; ref. to, 152, 153.
Tautochronous curve, 191.
Taylor, B., 242; ref. to, 234, 255.
Taylor s theorem, 243, 268, 269, 333,
342.
Tchebycheff, 368.
Tchirnhausen, 241; ref. to, 224, 226,
264, 328.
Tentative assumption, 75, 92. See
Kegula falsa.
Thales, 17, 18 ; ref. to, 16, 20, 21.
Thesetetus, 33; ref. to, 35, 36, 70.
Theodorus, 70 ; ref. to, 29.
Theodosius, 54; ref. to, 108, 125, 127.
Theon of Alexandria, 61 ; ref. to, 37,
51, 55, 65, 82.
Theon of Smyrna, 55, 58, 72.
Theory of equations. See Equations.
Theory of functions, 268, 269, 344, 345,
346, 347-362. See Functions.
Theory of numbers, 55, 76, 95, 108, 119,
131, 178-182, 252, 264, 280, 362-372.
Theory of substitutions, 329, 354.
INDEX.
421
Thermodynamics, 385, 398-401.
Theta-functions, 352, 353, 355, 380.
Theta-fuchsians, 345.
Theudius, 33.
Thomae, 353, 362.
Thome, 344 ; ref. to, 345.
Thomson, J., 385.
Thomson, J. J., 382; ref. to, 396, 397.
Thomson, Sir William, 395, 396 ; ref.
to, 283, 315, 358, 381, 382, 383, 388,
391, 393, 394, 395, 399, 400, 403. /
Kelvin (Lord).
Thomson s theorem, 359.
Three bodies, problem of, 253, 256, 377.
Thymaridas, 73.
Tides, 278, 383.
Timseus of Locri, 29.
Tisserand, 377.
Todhunter, 334; ref. to, L, XIII., 375.
Tonstall, 158.
Torricelli, 171.
Trajectories, 234, 238.
Triangulum characteristicum, 220.
Trigonometric series, 283, 339, 357. See
Fourier s series.
Trigonometry, 51, 56, 98-100, 109, 110,
115, 135, 140, 141, 154, 160, 161, 165,
238, 242, 245, 249, 259; spherical, 57,
* 115, 280, 294. .
Trisection of angles, 24, 31, 50, 153.
Trochoid, 171."
Trouton, 394.
Trudi, 324.
Tucker, Xm.
Twisted Cartesian, 312.
Tycho Brahe, 110, 139, 168.
Ubaldo, 183.
Ultimate multiplier, theory of, 379.
Ulug Beg, 114.
Undulatory theory of light, 192, 339,
379, 392-394.
Universities of Cologne, Leipzig, Ox
ford, Paris, and Prague, 136.
Valson, XIII.
Van Ceulen. See Ludolph.
Vandernionde, 278 ; ref. to, 264, 278.
Van Schooten, 189 ; ref. to, 190, 202.
Variation of arbitrary consonants, 378.
Varignon, 240 ; ref. to, 236.
Varying action, principle of, 292, 318,
379.
Venturi, 52.
Veronese, 305 ; ref. to, 307.
Versed sine, 99.
Vibrating rods, 387.
Vibrating strings, 242, 255, 262.
Vicat, 389; ref. to, 390.
Victorius, 79.
Vieta, 147; ref. to, 50, 142, 152, 153,
154, 167, 196, 202, 217, 264.
Vincent, Gregory St., 190, 197.
Virtual velocities, 34, 265.
Viviani, 172.
Vlacq, 165.
Voigt, XIIL, 365, 393.
Volaria, 237.
Von Helmholtz. See Helmholtz.
Von Staudt, 298, 299; ref. to, 292, 294,
295.
Vortex motion, 382.
Vortex rings, 382.
Voss, 306; ref. to, 336.
Waldo, 385.
Walker, 323.
Wallis, 192-195; ref. to, 98, 161, 177,
179, 187, 188, 197, 202, 229.
Waltershausen, XI.
Wand, 400.
Wantzel, 328.
Warring, 264, 330.
Watson, J. C., 377.
Watson, S., 341.
Wave theory. See Undulatory theory.
Waves, 382-385.
Weber, H. H., 355.
Weber, W. E., 394; ref. to, 357, 363,
388, 396, 398.
Weierstrass, 359 ; ref. to, 328, 339, 352,
353, 354, 359, 361, 362, 372.
Weigel, 219.
Weiler, 341.
Werner, 153.
Wertheim, 390.
Westergaard, 340.
Wheatstone, 386.
Whewell, IX., 43, 253.
Whiston, 216.
Whitney, 87.
Widmann, 150.
422
INDEX.
Wiener, XI.
Williams, 267.
Wilson, 264.
Wilson s theorem, 264.
Winds, 384-386.
Winkler, 392.
Witch of Agnesi, 260.
Wittstein, XII.
Woepcke, 83, 103.
Wolf, C., 241; ref. to, 167.
Wolf, R., XI.
Wolstenaolme, 341.
Woodhouse, 334.
Wren, 177; ref. to, 188, 198, 213, 287.
Wronski, 324.
Xenocrates, 29.
Xylander, 153.
Young, 392; ref. to, 386, 387.
Zag, 127.
Zahn, XII.
Zehfuss, 325.
Zeller, 365.
Zeno, 27.
Zenodorus, 51.
Zero (symbol for), 7, 88; origin of
term, 129.
Zeuthen, 313; ref. to, IX., 297.
Zeuxippus, 40.
Zolotareff, 371 ; ref. to, 372.
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